Given a Kirby diagram of a 4-manifold, what's its homotopy 2-type?

Yes: given a CW-complex $X$ the fundamental crossed module $\Pi_2(X,X^1)=(\partial \colon \pi_2(X,X^1) \to \pi_1(X^1))$, where $X^1$ is the 1-skeleton, represents the homotopy 2-type of $X$ (this can be stated in several ways). Moreover $\Pi_2(X,X^1)$ can be calculated combinatorially: $(\partial \colon \pi_2(X^2,X^1) \to \pi_1(X^1))$ is a totally free crossed module, and when you attach three handles you solely need to impose relations on $\pi_2(X^2,X^1)$ in order to get to $\pi_2(X,X^1)$

Now if you have a Kirby diagram (i.e. a handlebody decomposition), then squashing the handles along their core yields a CW-complex. So in theory the fundamental crossed module of the associated CW-complex can be calculated combinatorially. Except that it might be a non-trivial exercise to determine the attaching maps of the 3-handles in such that way that the relations on $\pi_2(X^2,X^1)$ become transparent. (For complements of knotted surfaces this is quite a doable thing.)

Some discussion is in J. Faria Martins, The fundamental crossed module of the complement of a knotted surface, Trans. Amer. Math. Soc. 361 (2009), no. 9, 4593–4630 https://arxiv.org/abs/0801.3921

and also in: Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry (https://arxiv.org/abs/1702.00868) section 3.4 and On 2-Dimensional Homotopy Invariants of Complements of Knotted Surfaces (https://arxiv.org/abs/math/0507239)

A step towards what you seem to be imagining may be in the following paper (available here):

J. Faria Martins, The fundamental crossed module of the complement of a knotted surface, Trans. Amer. Math. Soc. 361 (2009), no. 9, 4593–4630

As the title states this looks at the case of the complement of a knotted surface. I do not know if that helps.