Is constructing IND-CCA2 public key encryption schemes particularly easy with the KEM/DEM approach?
If I understand the question correctly, you are asking whether it's really needed to have the KEM be CCA secure, and maybe in the random oracle model it would suffice for it to just be an invertible one-way function. 

This would not be CCA2-secure. Specifically, let $f$ be any invertible one-way function, and construct $f'$ so that $f'(x)=0f(x)$ for every $x$. In addition, the inversion function works by ignoring the first bit of $y$ and then inverting the rest using $f^{-1}$. Note that $f'^{-1}(0y) = f^{-1}(1y)$ for every $y$. Thus, it is possible to change a single bit in the ciphertext, and the encrypted value remains untouched. Since the CCA2 experiment allows the adversary to query a decryption oracle on anything but the challenge ciphertext, this would allow it to obtain a decryption (by changing the appropriate bit of the ciphertext). I know that this is a stupid counterexample, but this is exactly the point of these counterexample: it demonstrates that additional properties are needed.

If we ask the same question for a trapdoor permutation then this seems more promising. The same KEM proof when using RSA (see Section 11.5.5) may go through in this case. I'm not guaranteeing it, since sometimes proofs fall on strange things, but it's my intuition. I suggest writing out a proof to make sure.