Even if you manage to statically link libc your program now has to be GPL/LGPL.
No, your program will not have to be GPL/LGPL. That is one of the common misconceptions about the LGPL. You are allowed to distribute a closed source proprietary software that statically links an LGPL-licensed library as long as you provide object files that allow users to re-link your closed sourced software to the library.
This is covered by the GPL FAQ: https://www.gnu.org/licenses/gpl-faq.html#LGPLStaticVsDynamic