I think I managed to solve this, but I have not verified this construction completely and rigorously. Still, I think it works and I am posting an answer at least for documentation and a more explicit verification later.

Compared to OP I will change notation slightly in that for this answer I am mostly dropping the notation associated with "finite order vector bundles", i.e. if $\rho:E\rightarrow J^r(\pi)$ is a vector bundle, let $\mathfrak G(\rho)$ denote the $\mathcal F_\infty(\pi)$-module of generalized sections, viz. smooth maps $\gamma:J^\infty(\pi)\rightarrow E$ that obey $\rho\circ\gamma=\pi^\infty_r$, and I will not use the pullback bundle $(\pi^\infty_r)^\ast E$ unless absolutely necessary.

Furthermore, another construction is useful. Fix an integer $r_0\in\mathbb N$ and suppose that for each $r\ge r_0$ we are given a vector bundle $\rho_r:E_r\rightarrow J^r(\pi)$, and for each $r\ge s$ we also have a vector bundle morphism $\eta^r_s:E_r\rightarrow E_s$ over $\pi^r_s$. Although not necessary to assume this, for applications it is safe to assume that $\eta^r_s$ is a fibration. Then the system $(E_r,\eta^r_s)$ of vector bundles is an inverse system and its projective limit $(E_\infty,\eta^\infty_r)$ is a vector bundle over $J^\infty(\pi)$. Such vector bundles will be called projective vector bundles and the "formal" smooth structure on $E_\infty$ is analogous to the one on $J^\infty(\pi)$. Then a smooth section of $E_\infty\rightarrow J^\infty(\pi)$ is equivalent to a sequence $(\gamma_{r_0},\gamma_{r_0+1},\dots)$ of generalized sections $\gamma_r\in\mathfrak G(\rho_r)$ such that for each $r\ge s$ we have $\gamma_s=\eta^r_s\circ\gamma_r$. This construction is completely analogous to that of the tangent bundle $TJ^\infty(\pi)$.

Let now $\rho:E\rightarrow J^s(\pi)$ be a vector bundle and $\rho_X:=\pi^s\circ\rho:E\rightarrow J^s(\pi)\rightarrow X$ be the composite bundle.

It can be seen that the $r$th prolongation $(\rho_X)^r:J^r(\rho_X)\rightarrow X$ factors through the fibered manifold $j^r\rho:J^r(\rho_X)\rightarrow J^r(\pi^s)$, where $J^r(\pi^s)=J^rJ^s(\pi)$ is a non-holonomic jet bundle, moreover, this is a vector bundle. Let $i_{r,s}:J^{r+s}(\pi)\rightarrow J^r(\pi^s)$ be the natural embedding of the $r+s$th holonomic jet space into the $(r,s)$ non-holonomic jet space, and let $$ \mathrm{pr}^r(\rho):\mathrm{Pr}^r(\rho)\rightarrow J^{r+s}(\pi),\quad\mathrm{Pr}^r(\rho)=(i_{r,s}^\ast J^r(\rho_X)) $$ be the pullback vector bundle to the holonomic jet space.

Given a section $\gamma\in\Gamma(\rho):J^s(\pi)\rightarrow E$, there is a unique section $\mathrm{pr}^r(\gamma):J^{r+s}(\pi)\rightarrow\mathrm{Pr}^r(\rho)$ which is given by $$ [\mathrm{pr}^r(\gamma)](j^{r+s}_x\phi)=j^r_x(\gamma\circ j^s\phi), $$ and we may call $\mathrm{pr}^r(\gamma)$ the $r$th prolongation of $\gamma$.

Prolongation is also well-defined for generalized sections. If $\gamma\in\mathfrak G(\rho)$ is a generalized section, then $\mathrm{pr}^r(\gamma)\in\mathfrak G(\mathrm{pr}^r(\rho))$ is the unique generalized section of $\mathrm{Pr}^r(\rho)\rightarrow J^{r+s}(\pi)$ that satisfies $$ [\mathrm{pr}^r(\gamma)]\circ j^\infty\phi=j^r(\gamma\circ j^\infty\phi) $$ for any local section $\phi\in\Gamma_\text{loc}(\pi)$.

For each $r\ge r^\prime$, $\mathrm{Pr}^r(\rho)$ projects onto $\mathrm{Pr}^{r^\prime}(\rho)$, so these "prolongation bundles" determine a projective vector bundle $\mathrm{Pr}(\rho)\rightarrow J^\infty(\pi)$, whose sections are the infinite prolongations $\mathrm{pr}(\gamma)=(\gamma,\mathrm{pr}^1(\gamma),\dots)$ of generalized sections.

Finally, let $\rho_E:E\rightarrow J^r(\pi)$ and $\rho_F:F\rightarrow J^s(\pi)$ be a pair of vector bundles. A total differential operator $\Delta:\mathfrak G(\rho_E)\rightarrow\mathfrak G(\rho_F)$ from (generalized sections of) $E$ to $F$ is an $\mathbb R$-linear operator such that there exists a smooth vector bundle morphism $\Delta_\sharp:\mathrm{Pr}(\rho_E)\rightarrow F$ (over $\pi^\infty_s$; smooth meaning it factors through some finite prolongation bundle) such that for any $\gamma\in\mathfrak G(\rho_E)$ we have $$ \Delta(\gamma)=\Delta_\sharp\circ\mathrm{pr}(\gamma). $$

I think I managed to solve this, but I have not verified this construction completely and rigorously. Still, I think it works and I am posting an answer at least for documentation and a more explicit verification later.

Compared to OP I will change notation slightly in that for this answer I am mostly dropping the notation associated with "finite order vector bundles", i.e. if $\rho:E\rightarrow J^r(\pi)$ is a vector bundle, let $\mathfrak G(\rho)$ denote the $\mathcal F_\infty(\pi)$-module of generalized sections, viz. smooth maps $\gamma:J^\infty(\pi)\rightarrow E$ that obey $\rho\circ\gamma=\pi^\infty_r$, and I will not use the pullback bundle $(\pi^\infty_r)^\ast E$ unless absolutely necessary.

Furthermore, another construction is useful. Fix an integer $r_0\in\mathbb N$ and suppose that for each $r\ge r_0$ we are given a vector bundle $\rho_r:E_r\rightarrow J^r(\pi)$, and for each $r\ge s$ we also have a vector bundle morphism $\eta^r_s:E_r\rightarrow E_s$ over $\pi^r_s$. Although not necessary to assume this, for applications it is safe to assume that $\eta^r_s$ is a fibration. Then the system $(E_r,\eta^r_s)$ of vector bundles is an inverse system and its projective limit $(E_\infty,\eta^\infty_r)$ is a vector bundle over $J^\infty(\pi)$. Such vector bundles will be called projective vector bundles and the "formal" smooth structure on $E_\infty$ is analogous to the one on $J^\infty(\pi)$. Then a smooth section of $E_\infty\rightarrow J^\infty(\pi)$ is equivalent to a sequence $(\gamma_{r_0},\gamma_{r_0+1},\dots)$ of generalized sections $\gamma_r\in\mathfrak G(\rho_r)$ such that for each $r\ge s$ we have $\gamma_s=\eta^r_s\circ\gamma_r$. This construction is completely analogous to that of the tangent bundle $TJ^\infty(\pi)$.

Let now $\rho:E\rightarrow J^s(\pi)$ be a vector bundle and $\rho_X:=\pi^s\circ\rho:E\rightarrow J^s(\pi)\rightarrow X$ be the composite bundle.

It can be seen that the $r$th prolongation $(\rho_X)^r:J^r(\rho_X)\rightarrow X$ factors through the fibered manifold $j^r\rho:J^r(\rho_X)\rightarrow J^r(\pi^s)$, where $J^r(\pi^s)=J^rJ^s(\pi)$ is a non-holonomic jet bundle, moreover, this is a vector bundle. Let $i_{r,s}:J^{r+s}(\pi)\rightarrow J^r(\pi^s)$ be the natural embedding of the $r+s$th holonomic jet space into the $(r,s)$ non-holonomic jet space, and let $$ \mathrm{pr}^r(\rho):\mathrm{Pr}^r(\rho)\rightarrow J^{r+s}(\pi),\quad\mathrm{Pr}^r(\rho)=(i_{r,s}^\ast J^r(\rho_X)) $$ be the pullback vector bundle to the holonomic jet space.

Given a section $\gamma\in\Gamma(\rho):J^s(\pi)\rightarrow E$, there is a unique section $\mathrm{pr}^r(\gamma):J^{r+s}(\pi)\rightarrow\mathrm{Pr}^r(\rho)$ which is given by $$ [\mathrm{pr}^r(\gamma)](j^{r+s}_x\phi)=j^r_x(\gamma\circ j^s\phi), $$ and we may call $\mathrm{pr}^r(\gamma)$ the $r$th prolongation of $\gamma$.

Prolongation is also well-defined for generalized sections. If $\gamma\in\mathfrak G(\rho)$ is a generalized section, then $\mathrm{pr}^r(\gamma)\in\mathfrak G(\mathrm{pr}^r(\rho))$ is the unique generalized section of $\mathrm{Pr}^r(\rho)\rightarrow J^{r+s}(\pi)$ that satisfies $$ [\mathrm{pr}^r(\gamma)]\circ j^\infty\phi=j^r(\gamma\circ j^\infty\phi) $$ for any local section $\phi\in\Gamma_\text{loc}(\pi)$.

For each $r\ge r^\prime$, $\mathrm{Pr}^r(\rho)$ projects onto $\mathrm{Pr}^{r^\prime}(\rho)$, so these "prolongation bundles" determine a projective vector bundle $\mathrm{Pr}(\rho)\rightarrow J^\infty(\pi)$, whose sections are the infinite prolongations $\mathrm{pr}(\gamma)=(\gamma,\mathrm{pr}^1(\gamma),\dots)$ of generalized sections.

Finally, let $\rho_E:E\rightarrow J^r(\pi)$ and $\rho_F:F\rightarrow J^s(\pi)$ be a pair of vector bundles. A total differential operator $\Delta:\mathfrak G(\rho_E)\rightarrow\mathfrak G(\rho_F)$ from (generalized sections of) $E$ to $F$ is an $\mathbb R$-linear operator such that there exists a smooth vector bundle morphism $\Delta_\sharp:\mathrm{Pr}(\rho_E)\rightarrow F$ (over $\pi^\infty_s$; smooth meaning it factors through some finite prolongation bundle) such that for any $\gamma\in\mathfrak G(\rho_E)$ we have $$ \Delta(\gamma)=\Delta_\sharp\circ\mathrm{pr}(\gamma). $$

Note that if $E\rightarrow J^r(\pi)=VY\rightarrow Y$, then this notion of prolongation reproduces the prolongation of evolutionary vector fields, but it does not reproduce the prolongation of non-vertical (generalized) vector fields over $Y$.