What you are calling Suslin measurable sets are also known as coanalytic sets (in the context of ZF + DC). The coanalytic sets are the sets obtained by applying your version of the Suslin operation to a family of open sets. One can prove in ZF + DC that the coanalytic sets are closed under your version of the Suslin operation, and hence every Suslin measurable set is coanalytic. This is not provable in ZF since as Gerald Edgar notes in the comments, it is consistent with ZF that every set is Suslin measurable, whereas ZF proves that there is a set that is not coanalytic (as will now be shown).

In fact, in ZF, one can show that there is a set with the Baire property that is not coanalytic. In fact, there is a surjection from $\omega^\omega$ onto the set of all coanalytic sets (first define a surjection onto the open sets, then onto the Suslin schemes, then onto the coanalytic sets). On the other hand, there is an injection from $P(\omega^\omega)$ to the set of subsets of $\omega^\omega$ with the Baire property: let $A\subseteq \omega^\omega$ be a meager set whose cardinality is the continuum (e.g., $A = 2^\omega$), and note that every subset of $A$ is meager and hence has the Baire property. Since there is no surjection from $\omega^\omega$ to $P(\omega^\omega)$, there is a set with the Baire property that is not coanalytic.

What you are calling Suslin measurable sets are also known as coanalytic sets (in the context of ZF + DC). The coanalytic sets are the sets obtained by applying your version of the Suslin operation to a family of open sets. One can prove in ZF + DC that the coanalytic sets are closed under your version of the Suslin operation, and hence every Suslin measurable set is coanalytic. This is not provable in ZF since as Gerald Edgar notes in the comments, it is consistent with ZF that every set is Suslin measurable, whereas ZF proves that there is a set that is not coanalytic (as will now be shown).

In fact, in ZF, one can show that there is a set with the Baire property that is not coanalytic. To see this, note that the set of all coanalytic subsets of $\omega^\omega$ is the surjective image of the continuum: ZF proves that the collection of open subsets of $\omega^\omega$ has cardinality $2^{\aleph_0}$, and so there are $(2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0}$ Suslin schemes, and hence the set of coanalytic subsets of $\omega^\omega$ is the surjective image of the continuum (being the surjective image of the Suslin schemes by definition). On the other hand, there is an injection from $P(\omega^\omega)$ to the set of subsets of $\omega^\omega$ with the Baire property: let $A\subseteq \omega^\omega$ be a meager set whose cardinality is the continuum (e.g., $A = 2^\omega$), and note that every subset of $A$ is meager and hence has the Baire property. Since there is no surjection from the continuum to $P(\omega^\omega)$, there must be a set with the Baire property that is not coanalytic.

What you are calling Suslin measurable sets are also known as coanalytic sets.

In ZF, one can show that there is a surjection from $\omega^\omega$ onto the set of all Suslin measurable sets (first define a surjection onto the open sets, then onto the Suslin schemes, then onto the Suslin measurable sets). On the other hand, there is an injection from $P(\omega^\omega)$ to the set of subsets of $\omega^\omega$ with the Baire property: let $A\subseteq \omega^\omega$ be a meager set whose cardinality is the continuum (e.g., $A = 2^\omega$), and note that every subset of $A$ is meager and hence has the Baire property. Since there is no surjection from $\omega^\omega$ to $P(\omega^\omega)$, there is a set with the Baire property that is not Suslin measurable.

What you are calling Suslin measurable sets are also known as coanalytic sets (in the context of ZF + DC). The coanalytic sets are the sets obtained by applying your version of the Suslin operation to a family of open sets. One can prove in ZF + DC that the coanalytic sets are closed under your version of the Suslin operation, and hence every Suslin measurable set is coanalytic. This is not provable in ZF since as Gerald Edgar notes in the comments, it is consistent with ZF that every set is Suslin measurable, whereas ZF proves that there is a set that is not coanalytic (as will now be shown).

In fact, in ZF, one can show that there is a set with the Baire property that is not coanalytic. In fact, there is a surjection from $\omega^\omega$ onto the set of all coanalytic sets (first define a surjection onto the open sets, then onto the Suslin schemes, then onto the coanalytic sets). On the other hand, there is an injection from $P(\omega^\omega)$ to the set of subsets of $\omega^\omega$ with the Baire property: let $A\subseteq \omega^\omega$ be a meager set whose cardinality is the continuum (e.g., $A = 2^\omega$), and note that every subset of $A$ is meager and hence has the Baire property. Since there is no surjection from $\omega^\omega$ to $P(\omega^\omega)$, there is a set with the Baire property that is not coanalytic.