What you want is not possible, but the problem could be the problem formulation itself. If you have a signal which is non-zero only at a finite number of frequencies, then your signal is a sum of sinusoids (or complex exponentials). What you can do is make their amplitudes equal, but this does not mean that $F(\omega_i)=\alpha$ (where $\omega_i$ are the frequencies of the sinusoids/complex exponentials), because the values $F(\omega_i)$ are not defined. This is because $F(\omega)$ is a sum of Dirac delta impulses, which are only defined by their integrals:
with some non-zero weights $w_i$. If your goal is to transform (1) into
with $\alpha$ being independent of the index $i$, then this is possible by filtering the signal $f(t)$ with a linear time-invariant filter whose frequency response satisfies
But note that this does not mean that $G(\omega_i)=\alpha$ because the values $G(\omega_i)$ are not defined.