MPIR is designed as a "drop-in replacement" for GMP (and it is a fork). Precisely the sorts of situations you envision can happen if you try to make them coexist.
It's probably an essential problem, as the vast majority of packages do not want a so-called "drop-in" replacement to require #including mpir.h and linking against libmpir.so (the default when building MPIR without the --enable-gmpcompat option).
Users typically want the library and .h file to be named compatibly, and they want the public interface to be the same. For a start, that is likely to clobber any existing gmp.h currently installed.
As far as the interface is concerned, the user facing, publicly documented interface should be the same as that of GMP (<= 5 at this stage). The internals are quite often different. Of course some projects link against internal, undocumented GMP functions.
My advice would be to stick with the official GNU package if you have concerns.
Bill.