Construction of invariants of 4-manifolds with the Kirby calculus

Disclaimer: Shameless self-advertising.

Yes, it can be done, and it's really beautiful! You can define the Crane-Yetter invariant with Kirby calculus, and possibly other TQFTs ("dichromatic models"). I've written this down in this article:

https://doi.org/10.1007/s00220-017-3012-9

If you want a version with more impressions and pictures, and less text, look at the talk slides:

https://www.manuelbaerenz.de/article/understanding-crane-yetter-model

The Broda invariant is a special case of the dichromatic framework, which was developed by Jerome Petit (and probably Alain Bruguières).

The dichromatic and Crane-Yetter invariants are stronger than signature and Euler characteristic. They are sensitive to the fundamental group (not just homology), but still fail to distinguish $S^2 \times S^2$ and $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$. They are probably still homotopy invariants, but this is a conjecture.

The Witten-Reshetikhin-Turaev approach to constructing quantum topological invariants of $3$-manifolds is to define them on framed links and to prove invariance under Kirby moves.

There is a paper of Broda which presents a $4$--manifold version of this strategy to construct "Witten-Reshetikhin-Turaev invariants" for $4$-manifolds.