Edited to more accurately address the OPs (restated) concerns:
It appears that one cannot hope for a general result for $L^p$, analytic (even entire) $F$ if $p \geq 2$. Consider, for example $$ f = \sum_{n = 1}^\infty n^n \chi_{[n, n + e^{-n^2}]}. $$ (By mollifying the cutoffs one could also take $f \in C^\infty$, but let me not worry about that for now.) We have $$ ||f||_{L^p}^p = \sum_{n = 1}^\infty n^{pn}e^{-n^2} < \infty $$ for all $1 \leq p < \infty$ (but $f \notin L^\infty$).
For $\xi \in \mathbb{C}$, define $$ F(\xi) = \int_{\mathbb{R}}f(y)e^{-i\xi y}dy. $$ From the definition of $f$, one immediately sees this is well-defined (the integral is absolutely convergent), and the usual differentiation under the integral sign (again, readily justified from the explicit form of $f$) shows that $F$ is an entire function. By the Hausdorff-Young inequality, $F \in L^p(\mathbb{R})$ for all $2 \leq p \leq \infty$, though we can also say with certitude that $F \notin L^1(\mathbb{R})$, since its Fourier transform $f$ is not $L^\infty$. By Carleson's theorem, $f(x) = \lim_{R \to \infty} \int_{-R}^R F(y)e^{ixy}dy$ for $x$-a.e., and of course $f$ is not of polynomial growth.