Poly-time reductions for proving EXPTIME-hardness are _not_ enough?
If your problem can be solved in time $O(2^k)$, where $k$&nbsp;is some number in the input, all you can say about its membership in conventional complexity classes is that it's in $\mathrm{2EXP}$, since an instance of length&nbsp;$\ell$ can be decided in time $O(2^{2^\ell})$.  You've not established that the problem is in&nbsp;$\mathrm{EXP}$.

$\mathrm{EXP}$, like all conventional complexity classes, is defined in terms of the length of the input, not the values represented in that input.