Conjecture: All FPT NP-complete languages are fixed-parameter-isomorphic
Serge Gaspers already mentioned why your conjecture is trivially true.

However, one can in fact get polynomial-time fixed-parameter isomorphisms,

which I now realize isn't that much less trivial, since it applies to every

ordered pair of non-trivial FPT problems with a reduction in the ordinary sense.



Let $c$ be an integer that's greater than the degree of the FPT algorithm for $\pi_1$,

and let $Y$ and $N$ be a yes and no instance respectively of $\pi_2$.

The following will be a polynomial-time fixed-parameter reduction from $\pi_1$ to $\pi_2$:

Try the FPT algorithm on $\pi_1$ for up to $n^{\hspace{.02 in}c}$ steps.

If that gives an answer, then output $Y$ or $N$ as indicated by that answer.

Otherwise, output the result of applying an ordinary polynomial-time reduction from $\pi_1$ to $\pi_2$.

Correctness and polynomial runtime are obvious. $\:$ Since $c$ is greater than the degree of the FPT algorithm for $\pi_1$, it's the case that for each fixed $k$, there are only finitely many input lengths $n$ for which the FPT algorithm's maximum runtime is not less than $n^{\hspace{.02 in}c}$. $\:$ Thus, for each fixed $k$, the above reduction has only finitely many outputs. $\:$ Therefore it satisfies your fixed-parameter condition.