But are the high levels of complexity associated with mathematical physics compatible with Occam's Razor?
The classical formulation of Occam's razor is: "Entia non sunt multiplicanda praeter necessitatem", or "one should not multiply things without a good reason".
Complex though many of the theories considered in mathematical physics are, their goal is nevertheless to find a description of the world that is simpler than alternative ways of explaining the same features of the world -- not in the superficial sense of being quicker to state, but in limiting the number of arbitrary choices that go into making a world. These theories may or may not eventually succeed in that goal, but simply because that is goal, the endeavor is perfectly compatible with Occam's Razor.
(Here, as everywhere, "simplicitly" is of course a matter of viewpoint, and open to discussion).
Complicated things require complicated answers. There are a lot of very different phenomena out there, and considering the relatively small number of assumptions made in mathematical physics, I think they do pretty well simplifying things. YMMV.
Occam's razor is not a concept as well defined as you usually assume. Or at least, it is relative to what looks parsimonious or simple for your particular taste. For instance, the simplest hypothesis to explain the universe would be: "every consistent structure that can exists, exists", and then you have the multiverse theory of Max Tegmark, in which our physical universe is just an example of an infinite number of possible mathematical structures. On the other side of the coin, you can see the simplicity not in the description, but in the number of objects and interactions assumed to exists. This is the example of modern physics, you have very complicated mathematical derivations, but in the end they describe a universe consisting of as few objects and interactions as possible (the complexity is in deriving the predictions, but it doesn't matter because you chose to apply the razor to the number of objects and interactions).
There is no fundamental contradiction between mathematical physics -- which is simply a rigorous way of formulating theory within the verified results discovered via experiment -- and Occam's razor.
But the question seems to stem from the idea that Occam's razor isn't precise enough to be serious. This is not really true, though -- Occam's razor is clearly required, for example in choosing the theory "general relativity" over "general relativity holds true until May 2027, then replaced with Newtonian gravity", even though both theories agree completely with experimental results so far.
Occam's razor can be made precise via Bayesian reasoning -- what the razor provides is a way to determine prior probabilities (more complicated theories have lower prior probability), then experimental results interfere with this probability distribution and produce a new one. This idea is formalised, e.g. in Solmonoff's theory of inductive inference (although I prefer the term interference, because you don't actually infer a definite conclusion, you just have experimental results to interfere with your probability distribution making it more and more precise and accurate), where complexity is formalised in terms of Kolmogorov complexity.