The philosophy here surely doesn't obligate one to be a Kantian, because we are talking about a model here.  Whatever lies behind the model, we realize that we have reduced it to terms we can handle.  There is no presumption of a deeper reality, just of a lack of perfect fit between the model and things we have not managed to observe.

I do believe that it stretches the notion of materialism past its breaking point, but not realism, or even 'physicalism' to the degree that we make no presumptions upon what is and what is not physical without experimental evidence.  The spirit of realistic reductivist monism remains, and we are not forced into accepting idealism by the fact that we treat mathematics as ideal.

Everyone in all the sciences has always done so, assuming the issues of reducing idealist mathematics to something consistent and realistic will be resolved inside mathematics.  Acting as though mathematics is an ideal domain does not mean you believe it is one, only that its issues are outside your purview.  For instance using continuous space does not imply real acceptance of the actual infinity of points, only acceptance that it is a good model, not worthy of breaking down farther.

The observation about fitting with relativity is unfair.  We have no idea whether or not relativity and quantum mechanics are actually compatible, but the idealization that the collapse of the wave function occurs simultaneously is just that, and idealization.  We have not yet come up with any way of testing whether this is information that travels at the speed of light, or slower, because we have not manage to find an unobserved frame large enough.  There is a good reason to think we can't, in that the odds of making or sustaining one become too small quite quickly.