Rust is definitely the worst language for this, writing curl in C++ would unlock memory and type safety while not throwing 30 years of language design out the window.
You really don't need to throw out all at once, you can just replace those components that profit most from rust guarantees, like where you parse external untrusted inputs first, with FFI one can then integrate that in the current code base without any calling overhead. Rinse repeat with next thing you profit most one, i.e., more like a Ship of Theseus process.
Leave it in C, or if you really have to, choose a safe language for the next 25 years. C++ ain't it.