Why is the lite version of Cramer-Shoup not IND-CCA2 secure?
CS-lite has the same homomorphic properties as ElGamal. In particular, compontentwise product of $\textsf{Enc}(pk,m_1)$ and $\textsf{Enc}(pk,m_2)$ is a valid encryption of $m_1 m_2$ (and hence will decrypt properly):

$$(g_1^r, g_2^r, A^r m_1, B^r) \cdot (g_1^s, g_2^s, A^s m_2, B^s) = (g_1^{r+s}, g_2^{r+s}, A^{r+s} (m_1m_2), B^{r+s})$$

So a simple CCA2 attack is to request $\textsf{Enc}(pk,m)$ for an unknown $m$, multiply this ciphertext by a fresh encryption of 1, and ask the decryption oracle to decrypt the result (revealing $m$).