Isn't there a way to record fees in the Customer Payments? Creating a Deposit transaction seems to be the only native option. Has someone created customization around this?

It's a little clunky, but the best option I've seen is to have your Payments go to Undeposited Funds instead of a bank account. Then you can do a deposit transaction and add/subtract other amounts and deposit it all in one shot.

NetSuite doesn’t handle this very elegantly, there are a couple of options assign the payment to the invoice , then journal the difference to a p/l bank charges account. Alternatively you could create a service item ' Bank charges' and post the sales to bank charges. Raise a credit memo using this item with the value of the bank charges this will then credit the A/R account with the value of the bank charges and debit p/l. The payment and the credit note could then be applied to the invoice to clear it completely.