I was wondering if the universal wave function can be thought of using a path integral approach? And if so, does each path correspond to a "branch" of the MWI of QM? Or have I misunderstood what is meant by a MWI branch?
Branches in the MWI are coarse-grained emergent things, and their number cannot be precisely quantified (see https://oxford.universitypressscholarship.com/view/10.1093/acprof:oso/9780199546961.001.0001/acprof-9780199546961-chapter-4 ). Depending on how finely you coarse-grain worlds and how the measurement is performed, more or less paths in the path integral may contribute significantly to each branch.
If you work only at the level of "individual" paths and define each path as a branch, like some sort of quantum Laplace's demon, then you must be very comfortable with some of your branches interfering with each other indefinitely rather than decohering into approximately separate worlds. By refusing to coarse-grain, you haven't really isolated any branching structure at all.
Yes, you can use the path-integral approach in the many-worlds formulation of QM but no, the paths of the path integral do not correspond to the branches of the many-worlds formulation of QM. Branching is associated with measurements. On the other hand, the path integral makes no reference to entanglement or decoherence -- the two crucial elements of a measurement process. As an extreme example, consider the following: in a world of a single particle with nothing else, there is no branching. However, obviously, you can still use path integral to calculate the time-evolution of the wavefunction of this particle.