I went to RBC yesterday to verify myself as I had opened account with them online. I was offered credit card ( which I politely refused as I already have one from Scotia and another from Amex is on the way). My PR card was not thoroughly scrutinized. Back side of the card was not checked for category code. I was verbally asked for my migration stream.
Would imagine many people had to show some proof of assets to qualify for a credit card if they were not working. Same for Canadians. Never hurts to contact other banks. They can only say no so no harm in trying.
This is the standard RBC gives to all regardless of the fact you're a PR because of refugee status or FSW. They also have some nice newcomer offers- I had got a 300 dollars paid in to my account. My spouse had 12 months waiver of all monthly fees.
Bottom line: RBC does a good job thus far as opposed to the other banks relative to newcomers.