Subexponential reduction

I am working on exact algorithms for an NP-hard problem $$P$$. I was able to get a $$(1.75^n$$) time algorithm for split graphs. When it comes to bipartite graphs, the problem becomes hard to tackle. Now, I want to prove that there is no sub-exponential algorithm for the problem on bipartite graphs. So to prove this, I have to provide a reduction from 3-SAT, which doesn't admit a $$2^{o(n)}$$ algorithm. I have reduced 3-SAT ($$m$$ clauses, $$n$$ varibales) to $$P$$ ($$G'$$ with $$n'$$ vertices) such that
(1) the reduction is done polynomial time.
(2) the size of the reduced instance $$n'$$ is $$m+n^3$$.
(3) 3-SAT is satisfiable if and only if $$G'$$ satisfy the necessary condition for the problem $$P$$
(4) $$G'$$ is bipartite
Now, can i say that the problem $$P$$ does not admit a $$2^{o(n)}$$ time algorithm for bipartite graphs? My main question is: does the reduction has anything to do with the size of the reduced instance, $$n'$$.

Under ETH, your problem cannot be solved in time $$2^{o(\sqrt[3]{n})}$$.