If you want to challenge someone to a competition, you could just say "Let's compete in <something>!" Or "Let's have a <something> contest!" More formally, swapping "Let's" out for "Let us" is possible, but that would be a little unusual. In either case, this is the sense of "I suggest that we […]".
If you want to ask someone to give you permission to compete, you would instead say "[Will you] let us compete, please?" (And presumably give some sort of reason why.) In this case, you cannot use the contracted form "let's" at all, because you're emphasizing the allowing sense of "let", and that is never what "let's" means.
"Let us" (in either sense) can also be changed to "let them" if you don't include yourself.
Neither "do" or "make" are really appropriate here at all, without major revisions to try to fit them in somewhere. And "let" without "us" or "them" immediately following won't work either.