It's possible I'm over-thinking this:
If $f \in H(1)$, let
$$ f_\delta(z) = \frac{1}{\sqrt{\pi\delta}} \int_{(0)} f(w) \exp ((w-z)^2/\delta) dw, $$
then
$$ f_\delta(x+iy) = \frac{i\exp (x^2/\delta)}{\sqrt{\pi\delta}} \int_{-\infty}^\infty f(i(t+y)) e^{-2itx/\delta} \exp (-t^2/\delta) dt, $$
so $f_\delta$ is entire (integral converges rapidly for all $z$), and $f \in H(b)$ for all $b>1$, by interchanging integrals in the norm and using
$$ (1+|t+y|) \le (1+|t|)(1+|y|). $$
Also, on $S_a$,
$$ f_\delta(z) = \frac{1}{\sqrt{\pi\delta}} \int_{(0)} f(z+w) \exp z^2 dw, $$
by contour shifting.
Then $f_\delta \to f$ as $\delta \to 0$, by the usual approximation to the identity argument.

Maybe the norm should really be
$$ ||f||^2_a:=\sup_{|\sigma|<a} \int_{-\infty}^\infty |f(\sigma+it)|^2 (1+|t|)^{100} dt \, ? $$