I have paid money into my Co-Op bank account Smile at the Post Office landside in Terminal 3. I did it on a Sunday too, which at the time was an unusual thing to do (these days lots of post offices seem to be open on a Sunday).

Details from the HAL website here.

HSBC is listed on the Post Office site as accepting cash and cheque deposits, as well as cash withdrawals free of charge.

If you arrive early you can do a Flight Connections transfer out to T3, but then you would have to go landside back to T5.