Want to do mathematical physics - if first choice dept not a good fit, can I switch?
It seems like "you" should do a lot more research into existing graduate programs, since you haven't mentioned maybe the most obvious thing you should do: apply to an applied math Ph.D. program.  You should look carefully at what different programs expect, since "applied math" is a slippery concept; however, as such, applied math programs don't have an uniform an expectation of student background as pure math or physics programs.  Interdisciplinary programs are hot at the moment, but unlike a usual math or physics degree, what's on offer will vary dramatically from university to university.  In my new hometown of Waterloo alone, there are (I think usually funded) 


Masters in Applied Math (https://uwaterloo.ca/applied-mathematics/graduate-students/applying), 
Perimeter Scholars International (https://www.perimeterinstitute.ca/training/about-psi), and 
a special stream in the Institute for Quantum Computing (https://uwaterloo.ca/institute-for-quantum-computing/programs/graduate-studies).


That's a bit out of the norm, but there are many other programs out there.