/proc is a pseudo-filesystem supplied by the Linux kernel, so as to provide an interface to applications to learn about system runtime information (running processes, mounts, memory, etc.). Therefore there's no way to "mount" it as R/W since that's meaningless.
Some entry points (NOT files), however, are writable by themselves, for example most entries under
/proc/sys. Do note that by writing to these entries you're not writing any files, but actually changing system parameters.
If your program is trying to write to another non-writable entry in
/proc, it's probably malformed and should be tweaked. Again,
/proc is not a real filesystem and everything inside are interfaces from the Linux kernel.
Read more (related or similar):
Documentations (usually very long and complete and unsuitable for casual read):