Is there any sort of benefit to be gained by porting the code to C++?
I think the key phrase here is "porting". What do you mean by that? If you truly port the code to C++, make use of C++' features like templates etc., there is a chance this might speed up the code. (For example, C++'
std::sort() is known to be faster than C's
qsort() because a C++ compiler can inline the calls to the comparison function.) OTOH, the algorithms used are likely to be tailored to C-style programming, so it might well be that the only way to get a measurable speed gain would be to start from scratch and implement and polish for years to make the library shine in C++. However, a C++ compiler can do more compile-time checks for code written in C++ style, so you would gain there.
But if you only make the C code compile with a C++ compiler, the task is questionable. Given C code, a C++ compiler cannot make (m)any assumptions a C compiler cannot make to base its optimizations on. And C-style code is prone to shutting up the compiler (hard casts, no templates, etc.), so you are unlikely to gain any improved type safety at compile time.
What are the downsides that might cancel out these benefits?
Either way, you need to change the code. That will introduce bugs and incompatibilities. That alone is a big downside that will never be outweighed by the meager chances for improvements listed above.