Is a Bank required to withhold and remit if the customer wants us to? Operations wants to tell customers that they will have to do submit their own withholding. I've never heard of this before.

You are required to withhold and remit unless the customer tells you not to by signing a properly completed W4-P or its equivalent. Your bank is liable for the amount it should have withheld but did not, regardless of whether you still have the funds on deposit.

Requiring customers to file a written election against withholding sounds a bit lazy...