Cashier's checks payable to a third party - endorsed over to our customer deposited at ATM.

We placed a hold for 7 days using reason as "Other" - 3rd party check. Person placing hold did not think that the endorsement of our customer matched what we had on our signature card.

I thought we should have made $200 available next day and remainder on 2nd business day.

Thoughts please.

Case by case holds require $200 next business day. Exception holds do not. (Note if we combine a case by case and large item exception, then $200 is required next business day and $4,800 the second business day.) A cashier's check loses its mandatory next business day availability if it is not deposited in person into the account of the named payee.

The point is moot. Since the payee of the check has up to three years depending on your state's Uniform Commercial Code to claim forged endorsement, a 7 business day hold leaves you with approximately 1085 days of potential liability exposure (give or take depending on holidays and leap year.) If we were concerned enough to place a hold, maybe we should have refused the deposit.

Can the bank refuse the deposit of cashiers checks (or any check) when deposited at the ATM when it is reviewed by bank staff after the deposit has been made?

Yes, you can. Just mail it back to the customer with an explanation.