Random semantic security and bijective PRGs
There are a few fundamental problems with your approach:


When you use the construction $m \mapsto E(k,G(m))$ in the context of the standard CPA game, you are feeding adversarially chosen values into the PRG $G$. PRG security does not guarantee anything about PRG outputs in this setting. PRG security can only be applied when the PRG input is uniformly chosen, and used nowhere else in the system (in particular, it can't be known to / chosen by the adversary).
As you mention, decryption requires $G$ to have an efficiently computable inverse, which is incompatible with PRG security. The alternative is that $G$ has no stretch, which basically defeats the purpose of a PRG. If you relax the stretch requirement of a PRG, then even the identity function is a secure PRG in the sense of its outputs looking random. Clearly when you use the identity function in your construction, you haven't actually done anything to $\mathcal{E}$.



  Is there a different approach to building $\mathcal E′$ that gets around this potential security flaw?


Yes, but I don't know how to give a reasonable hint without trivializing the problem. You don't need to introduce any additional crypto primitives beyond the original scheme $\mathcal{E}$.