Consider a subsequence $\,g_n\,$ of $\,f_n\,$ which is uniformly convergent. Then consider a subsequence $\,h_n\,$ of $\,g_n\,$ such that sequence $\,h^{-1}_n\,$ is uniformly convergent. Then the limit function $\,h\,$ of $\,h_n\,$ is bilinear with the bi-constant equal the same $\,c.$   Great

REMARK This proves what @NikWeaver has already said in his comment under the OP's Question. (Yes, I agree with the comment by @NatEldredge).

Consider a subsequence $\,g_n\,$ of $\,f_n\,$ which is uniformly convergent. Then consider a subsequence $\,h_n\,$ of $\,g_n\,$ such that sequence $\,h^{-1}_n\,$ is uniformly convergent. Then the limit function $\,h\,$ of $\,h_n\,$ is bilipschitz with the bi-constant equal the same $\,c.$   Great

REMARK This proves what @NikWeaver has already said in his comment under the OP's Question. (Yes, I agree with the comment by @NatEldredge).