For the adjusted question, take a group G with generators $x_n$ and relations $x_n=1$ if the statement with Gödel number n is provable. (This is recursively presented because you can enumerate all proofs.) Higman embed G into an fp group H. Take a Gödel number m of a statement which is true but not provable (exists by incompleteness). We cannot give a finite proof that $x_m\neq 1$ in H.

For the adjusted question, take a group G with generators $x_n$ and relations $x_n=1$ if the statement with Gödel number n is provable. (This is recursively presented because you can enumerate all proofs.) Higman embed G into an fp group H. Take a Gödel number m of a statement which is true but not provable (exists by incompleteness). We cannot give a finite proof that $x_m\neq 1$ in H.

edit Following Joel's kind suggestion I should use Rosser sentences instead of Gödel sentences to be independent of the background meta theory.