Specifically the expensive part is getting a C89 port of mrustc to compile rustc 1.90.0. mrustc needs C++ right now, bootstrapping via TCC by writing it in C89 would be the shorter chain.
I have no doubt that a Rust compiler can be implemented in C. But it's a different project to write one in C from start than trying to migrate an existing C++14 based project to C. The former is likely "easier". An alternative approach could be to migrate the C++14 code back to C++98 and get rid of the exceptions, which makes it transpilable to C (using the cfront approach). Though the result might not be maintainable C, but still C.
EDIT: if you manage to migrate it back to a sufficiently moderate C++11 subset, or even C++98, you could use GCC 4.7.4 to compile it, which itself is written in C.