Here are some remarks and pointers to the literature which are too long for a comment.

The map $$ \mathsf{Dist}(X_1) \times_{\mathsf{Dist}(X)} \mathsf{Dist}(X_2) \longrightarrow \mathsf{Dist}(X_1 \times_X X_2) $$ is a well-known construction called conditional product. While this has been rediscovered many times over, the first axiomatics that I am aware of were proposed in a paper by Dawid and Studený. More recently, one categorical axiomatization based on the resemblance with sheaf-theoretic gluing was given in this paper. Another categorical axiomatization has been given by Simpson in this paper, where conditional products are called local independent products. Note that the first two references only consider conditional products of the special type $\eqref{laxproduct_special}$, but I believe this to be only a minor restriction.

An important caveat is that although the conditional product is natural in the left and right arguments, it is not natural in the middle one. This plays a prominent role in such categorical axiomatizations.

A general construction of conditional products for suitable monads can be extracted from my latest paper. As described in Section 3, the Kleisli category of any symmetric monoidal affine monad $M$ on a category with finite products is a Markov category. If this Markov category has conditionals in the sense of Definition 11.1, then Definition 12.8 shows that one can form conditional products in the special form $$\tag{1}\label{laxproduct_special} M(A \times B) \times_{M(B)} M(B \times C) \longrightarrow M(A \times B \times C), $$ and I suspect that one can arrive at the general form as in the OP with a bit more fiddling. However, I do not know how to translate the assumption for the Kleisli category to have conditionals into an assumption on the monad, and this may end up being related to your $f^M$. I also do not know whether the existence of conditionals is related to any sort of "noncancellativity"; I have investigated one such condition in Definition 11.19 and after, but it does not seem to play any role in the context of conditional products. (By the way, what is called "noncancellativity" in the OP should arguably be called positivity or zerosumfreeness, since cancellativity usually means more something like $a + c = b + c \: \Rightarrow \: a = b$.)

Finally, I agree with Todd Trimble that the question also triggers strong associations with weakly cartesian monads. Replacing the title of the second column by "has weakly cartesian underlying functor and weakly cartesian multiplication" indeed gives the exact same yes/no answers. This can be seen by applying the results of this paper by Clementino, Hofmann and Janelidze, where this property is called condition (BC). Properties like zerosumfreeness—i.e. what the OP calls "noncancellativity"—indeed play a major role in characterizing monads on $\mathsf{Set}$ which satisfy condition (BC).

Here are some remarks and pointers to the literature which are too long for a comment.

The map $$ \mathsf{Dist}(X_1) \times_{\mathsf{Dist}(X)} \mathsf{Dist}(X_2) \longrightarrow \mathsf{Dist}(X_1 \times_X X_2) $$ is a well-known construction called conditional product. While this has been rediscovered many times over, the first axiomatics that I am aware of were proposed in a paper by Dawid and Studený. More recently, one categorical axiomatization based on the resemblance with sheaf-theoretic gluing was given in this paper. Another categorical axiomatization has been given by Simpson in this paper, where conditional products are called local independent products. Note that the first two references only consider conditional products of the special type $\eqref{laxproduct_special}$ below, but I believe this to be only a minor restriction.

An important caveat is that although the conditional product is natural in the left and right arguments, it is not natural in the middle one. This plays a prominent role in such categorical axiomatizations.

A general construction of conditional products for suitable monads can be extracted from my latest paper. As described in Section 3, the Kleisli category of any symmetric monoidal affine monad $M$ on a category with finite products is a Markov category. If this Markov category has conditionals in the sense of Definition 11.1, then Definition 12.8 shows that one can form conditional products in the special form $$\tag{1}\label{laxproduct_special} M(A \times B) \times_{M(B)} M(B \times C) \longrightarrow M(A \times B \times C), $$ and I suspect that one can arrive at the general form as in the OP with a bit more fiddling. However, I do not know how to translate the assumption for the Kleisli category to have conditionals into an assumption on the monad, and this may end up being related to your $f^M$. I also do not know whether the existence of conditionals is related to any sort of "noncancellativity"; I have investigated one such condition in Definition 11.19 and after, but it does not seem to play any role in the context of conditional products. (By the way, what is called "noncancellativity" in the OP should arguably be called positivity or zerosumfreeness, since cancellativity usually means more something like $a + c = b + c \: \Rightarrow \: a = b$.)

Finally, I agree with Todd Trimble that the question also triggers strong associations with weakly cartesian monads. Replacing the title of the second column by "has weakly cartesian underlying functor and weakly cartesian multiplication" indeed gives the exact same yes/no answers. This can be seen by applying the results of this paper by Clementino, Hofmann and Janelidze, where this property is called condition (BC). Properties like zerosumfreeness—i.e. what the OP calls "noncancellativity"—indeed play a major role in characterizing monads on $\mathsf{Set}$ which satisfy condition (BC).

Here are some remarks and pointers to the literature which are too long for a comment.

The map $$ \mathsf{Dist}(X_1) \times_{\mathsf{Dist}(X)} \mathsf{Dist}(X_2) \longrightarrow \mathsf{Dist}(X_1 \times_X X_2) $$ is a well-known construction called conditional product. While this has been rediscovered many times over, the first axiomatics that I am aware of were proposed in a paper by Dawid and Studený. More recently, one categorical axiomatization based on the resemblance with sheaf-theoretic gluing was given in this paper. Another categorical axiomatization has been given by Simpson in this paper, where conditional products are called local independent products.

An important caveat is that although the conditional product is natural in the left and right arguments, it is not natural in the middle one. This plays a prominent role in such categorical axiomatizations.

A general construction of conditional products for suitable monads can be extracted from my latest paper. As described in Section 3, the Kleisli category of any symmetric monoidal affine monad $M$ on a category with finite products is a Markov category. If this Markov category has conditionals in the sense of Definition 11.1, then Definition 12.8 shows that one can form conditional products in the special form $$ M(A \times B) \times_{M(B)} M(B \times C) \longrightarrow M(A \times B \times C), $$ and I suspect that one can arrive at the general form as in the OP with a bit more fiddling. However, I do not know how to translate the assumption for the Kleisli category to have conditionals into an assumption on the monad, and this may end up being related to your $f^M$. I also do not know whether the existence of conditionals is related to any sort of "noncancellativity"; I have investigated one such condition in Definition 11.19 and after, but it does not seem to play any role in the context of conditional products. (By the way, what is called "noncancellativity" in the OP should arguably be called positivity or zerosumfreeness, since cancellativity usually means more something like $a + c = b + c \: \Rightarrow \: a = b$.)

Finally, I agree with Todd Trimble that the question also triggers strong associations with weakly cartesian monads. Replacing the title of the second column by "has weakly cartesian underlying functor and weakly cartesian multiplication" indeed gives the exact same yes/no answers. This can be seen by applying the results of this paper by Clementino, Hofmann and Janelidze, where this property is called condition (BC). Properties like zerosumfreeness—i.e. what the OP calls "noncancellativity"—indeed play a major role in characterizing monads on $\mathsf{Set}$ which satisfy condition (BC).

Here are some remarks and pointers to the literature which are too long for a comment.

The map $$ \mathsf{Dist}(X_1) \times_{\mathsf{Dist}(X)} \mathsf{Dist}(X_2) \longrightarrow \mathsf{Dist}(X_1 \times_X X_2) $$ is a well-known construction called conditional product. While this has been rediscovered many times over, the first axiomatics that I am aware of were proposed in a paper by Dawid and Studený. More recently, one categorical axiomatization based on the resemblance with sheaf-theoretic gluing was given in this paper. Another categorical axiomatization has been given by Simpson in this paper, where conditional products are called local independent products. Note that the first two references only consider conditional products of the special type $\eqref{laxproduct_special}$, but I believe this to be only a minor restriction.

An important caveat is that although the conditional product is natural in the left and right arguments, it is not natural in the middle one. This plays a prominent role in such categorical axiomatizations.

A general construction of conditional products for suitable monads can be extracted from my latest paper. As described in Section 3, the Kleisli category of any symmetric monoidal affine monad $M$ on a category with finite products is a Markov category. If this Markov category has conditionals in the sense of Definition 11.1, then Definition 12.8 shows that one can form conditional products in the special form $$\tag{1}\label{laxproduct_special} M(A \times B) \times_{M(B)} M(B \times C) \longrightarrow M(A \times B \times C), $$ and I suspect that one can arrive at the general form as in the OP with a bit more fiddling. However, I do not know how to translate the assumption for the Kleisli category to have conditionals into an assumption on the monad, and this may end up being related to your $f^M$. I also do not know whether the existence of conditionals is related to any sort of "noncancellativity"; I have investigated one such condition in Definition 11.19 and after, but it does not seem to play any role in the context of conditional products. (By the way, what is called "noncancellativity" in the OP should arguably be called positivity or zerosumfreeness, since cancellativity usually means more something like $a + c = b + c \: \Rightarrow \: a = b$.)

Finally, I agree with Todd Trimble that the question also triggers strong associations with weakly cartesian monads. Replacing the title of the second column by "has weakly cartesian underlying functor and weakly cartesian multiplication" indeed gives the exact same yes/no answers. This can be seen by applying the results of this paper by Clementino, Hofmann and Janelidze, where this property is called condition (BC). Properties like zerosumfreeness—i.e. what the OP calls "noncancellativity"—indeed play a major role in characterizing monads on $\mathsf{Set}$ which satisfy condition (BC).