We had a customer deposit a Cashiers Check that was eventually charged back to the account, but by the time the check was charged back, the customer figured out that it was an internet scam and he had already moneygramed over $3000.00 of the deposit. Does the bank need to complete a SAR?

I know my boss would require me to file one.

Yes - if:
1) Deposited Cashiers Check was more than $5,000.
2) You honestly believe the customer was aware that the deposited item was fraudulent/worthless when he/she transferred a portion out of the account.

This would be an intentional attempt to defraud the bank.