AES-CBC and IV -- Encrypting multiple blocks of a file with the same IV?
It is insecure to reuse the $IV$ with AES-CBC. At the very least, if the files have a common prefix, this will be revealed as a common prefix of the ciphertexts. For AES-CBC, the only way to ensure confidentiality is to use random $IV$s.

However, if you are not restricted to a particular CBC mode, the nonce-based Counter mode (CTR) might solve your problem. Each block is then encrypted with unique number (nonce) $N$ instead of $IV$. It does not have to be secret, moreover, it can be predictable (for example, it can be the block index).

Another option is to try modes of operation dedicated to small-block encryption.  XTS, which employs key-dependent tweaks instead of IVs and nonces, is a widely used example.