The bank must be one that has a relationship with the Payment Gateway, where in the case of T&T it is most often the First Atlantic Commerce gateway.
The T&T Online Payments Scene
Arvinda R

Can you explain why this is? Compared to Paypal, First Atlantic Commerce is expensive.

First Atlantic Commerce — FAQ
Why Is There No Mention Of Pricing On Your Web Site?

Our solutions are priced on a per-merchant basis, tailored specifically to the services you require. There are a range of variables that range from business type, to products being sold, existing and future processing volumes, currencies required, chargeback history and so on.

How Often Do I Pay Bank Processing Fees?

Banks take their fees direct from merchant processing each day and in line with established contractual procedures.

How Often Do I Pay Payment Gateway Fees?

Our payment gateway fees should be paid monthly and on receipt of invoice.

Read it yourself here:

In my opinion people are just using FAC because other people are using it. Accepting TT currency is not a valid argument in my opinion.

Paypal only has a $0.30 USD transaction fee in addition to their processing fee. No monthly fees and no setup fees.

If you online store makes $0 you pay $0 with PayPay. With FAC when you set up your shop you pay, when you make a transaction you pay, processing your pay, every month you pay weather you make $0 or not.