CBC first block malleability
Yes, you can modify the initialization vector to accomplish your goal (as mentioned in the comments, the initialization vector is the first block of the cipher text). So, by computing a new initialization vector, $IV_2 = IV_1 \oplus x$, the cipertext/IV pair $(c_1, IV_2)$ is a valid encryption of $m_2$ with CBC mode. This is because $m_2[1] = m_1[1] \oplus x$, so 
$$m_2[1] \oplus IV_2 = (m_1[1] \oplus x) \oplus (IV_1 \oplus x) = m_1[1] \oplus IV_1$$ This way, the first blocks of input to the blockcipher are the same for both messages ($\oplus$'d with their respective IVs), and the encrypted outputs are valid for both messages.