It is a bit strange that you are talking about reducing SUBSET-SUM to 3SAT to conclude $P \neq NP$ from the hypothesis of non-existence of a sub-exponential algorithm for SUBSET-SUM.
The fact that SUBSET-SUM is in $NP$, coupled with the hypothesis that it has no polynomial time algorithms is enough to show that $P \neq NP$. There is no need to talk about any reductions, or even pick problems which are $NP$-Complete.
So let us try to re-interpret your problem to what I think you might be asking:
Suppose problem $A$ can be reduced to problem $B$ in polynomial time.
Does the non-existence of a sub-exponential time algorithm (
$o(c^n)$ for every $c \gt 1$, note: smallOh) for $A$ imply the non-existence of a
sub-exponential time algorithm for $B$?
The answer is not necessarily.
This is because the polynomial time algorithm for reduction can blow up the size of your input!
Suppose every reduction from $A$ to $B$ changed the input size from $n$ to $n^3$.
Now any $\Theta(2^{\sqrt{k}})$ time algorithm for $B$ (with input size $k$) will solve $A$ (with input size $n$) in $\Theta(2^{n^{3/2}})$ time, and does not contradict the non-existence of a sub-exponential time algorithm for $A$.
See also this question on cstheory.SE.