At first glance, I thought the obvious answer to that question was "Nope..that's not legal".

But then I pulled up an application to a BoA card



They offer non-resident aliens an oppurtunity to get the card as well.
So surely if OP is getting to hold on to his SSN and will have proof of residence, he'd be able to get a card.

As to what happens if the CC provider demands a 4605-T, I've no idea.