Is a bank ever required to complete a CMIR if the bank itself is sending or receiving the the currency or monetary instrument? Or is the bank always exempt?

"Exempt" is not quite the right word.

The filing requirement just does not apply to banks when it's the customer who moved the money; 1988 - R002.

So if, for example, a foreign loan customer mails us a check for over $10,000 to be applied as a loan payment. It is still on the customer?

From the instructions:

Exceptions: Reports are not required to be filed by:
(1) a Federal Reserve bank,
(2) a bank, a foreign bank, or a broker or dealer in securities in respect to currency or other monetary instruments mailed or shipped through the postal service or by common carrier
[abridged]

Thank you.