My Chinese is not so good, but it looks as if they are refusing you a credit card, not a debit card.
You want a VISA 金融卡, (jin rong ka) or debit card.
Who wrote you that message above, your branch or the credit card department?
There is a TBB near my house so I popped in today for a minute to ask them about your situation. The woman there said that a debit card would be no problem and that you could just pop into another branch and apply for it. They do have a no credit card for foreigners rule from what she told me.
The branch was in Kaohsiung, I'm not sure where you live however. Just pop into another branch of TBB from what it looks like, don't mention the previous conversation with your branch, and ask for the Jin Rong Ka. Hopefully someone there will know what they are doing.
If they are still refusing you, just sent a letter or email to the bank (you can find the address online) spelling out your displeasure at having being refused service, which necessitates you having to take out all your money on payday and move it to a real bank. That might move things along.
*However it looks like your branch is just simply misinformed. PM me if you need any further help in this matter.