There are two questions.
For the specific functional equation considered in this question http://mathoverflow.net/questions/114875/, the formula I gave covers all entire solutions. I added the reference references there.
On the general question about "Fourier transform" of entire functions or functions on the real line which are not in $L^p$. One usually replaces Fourier transform with various versions of Laplace transform. There are many versions, for various problems. I recommend Hormander, Analysis of differential operators..., Chap. 9, or the paper MR0199747 Ljubič, Ju. I.; Tkačenko, V. A. Theory and certain applications of the local Laplace transform. (Russian) Mat. Sb. (N.S.) 70 (112) 1966 416–437. There is an English translation in Math USSR Sbornik. There is also a nice little book by Carleman of Fourier transform (in French).
Edit. See also http://mathoverflow.net/questions/31458 for an example how Laplace transform of entire functions is used.