I want to be careful with Cap One (I have CC and Auto Loan with them), they've been pretty nice to me.
If you're not opted out, doesn't Chase/Citi have prequalifier links up, and running now, as well as Discover?
I have received pre-approval letters from Discover, Chase and Wells Fargo. I was a authprized user on a card from Citibank that was not paid off.
Maybe your best bet is to bite the bullet, and app for what you want. If you get a denial, you'll have the reasoning behind the decision, and can recon from there.