The Borel sets and also the analytic sets are known to be Lebesgue Measurable in $ZFC$. The Analytic sets were defined by Suslin using the Suslin operation or the $A$-operation and Luzin proved that the Lebesgue measurable sets are closed under this operation. So the closure of the analytic sets under the Suslin operation is a class, that extends the analytic sets, in which all the sets are Lebesgue measurable.
However the sets $\Delta_2^1$ (The second level of the projective hierarchy) cannot be all be shown to be Lebesgue measurable in $ZFC$ (by Gödel's results).
Are there any known result concerning this "cap" of sets that can be proven to be Lebesgue measurable in $ZFC$?
R. Solovay proved that the provably $\mathbf\Delta^1_2$ sets are Lebesgue measurable (and have the property of Baire). A set $A$ is provably $\mathbf\Delta^1_2$ iff there is a real $a$, a $\Sigma^1_2$ formula $\phi(x,y)$ and a $\Pi^1_2$ formula $\psi(x,y)$ such that $$A=\{t\mid \phi(t,a)\}=\{t\mid\psi(t,a)\},$$ and $\mathsf{ZFC}$ proves that $\phi$ and $\psi$ are equivalent. This readily follows from his arguments in his paper on the consistency of all sets of reals being measurable.
A. Kanamori's book has details in section 14.
As you mentioned, we cannot remove the provability requirement (since, by K. Gödel's arguments, for any real $a$, $L[a]$ is a model with a nonmeasurable $\Delta^1_2(a)$ set). On the other hand, without going beyond $\mathsf{ZFC}$ in consistency strength, larger pointclasses can be shown measurable under suitable additional axioms, such as $\mathsf{MA}$:
From $\mathsf{MA}$ (plus $\lnot\mathsf{CH}$) it follows that every $\mathbf\Sigma^1_2$ set of reals is Lebesgue measurable and has the property of Baire. This is already established in the original Martin-Solovay paper.
For a modern exposition I suggest the Bartoszyński-Judah book.
Martin's axiom can be forced over any model of $\mathsf{ZFC}$, so it is equiconsistent with $\mathsf{ZFC}$. Even further, S. Shelah showed that $\mathsf{ZFC}$ is equiconsistent with the measurability of $\mathbf\Delta^1_3$ sets, and Judah strengthened this fact by showing that adding $\aleph_1$ random reals to a model of $\mathsf{MA}$ results in a model where all $\mathbf\Delta^1_3$ sets are Lebesgue measurable (this is described in the Bartoszyński-Judah book as well).
Nevertheless, you cannot go much further by restricting to $\mathsf{ZFC}$ consistencywise: Shelah showed that the measurability of the $\mathbf\Sigma^1_3$ sets implies the existence of inaccessible cardinals in $L$.