I'd do it the other way round. A VM is a bit of a pain and you can't target a constant-time-only VM with a regular C compiler. Instead I'd produce a restricted C compiler that only accepted constant time programs, and mapped directly to a limited set of machine instructions. Regardless of what you do, you'd have to produce a whilelist of constant-time machine instructions, so you might as well emit them directly rather than relying on a general purpose optimising compiler to do the right thing.

You can then emit an object file containing the functions you're interested in and link it to the rest of your non-constant-time program.