I decided to try for an AmEx, and they couldn't instantly approve and are mailing a decision. I've had that go both ways as far as getting approved. The last two (a year ago) both got approved. Annnnd, due to my lack of patience, I went ahead and applied for a Discover as well. I got instant approval for a fairly decent amount. Whether I come out ahead on my credit score in the end, I've now got some windfall financially for NYC.
Ironic confession time: My oldest card was a tiny limit card with an annual fee. I called the bank to see, since I've had this card since 2008 and neeeever been late on a payment including the other two cards I've had with this same bank, if they would waive the fee or convert it to a different type of card. They said they couldn't without losing the credit history. So it got put out of my mind, and I forgot to make the payment on the fee (I don't use this card much anymore).

I called the bank (again) to see if they would waive the late fee and not report to the credit bureau since I've neeeever been late on a payment, and I was only 1 day late, and he told me that it was all good - my type of card doesn't charge a late fee on a balance under $100, and doesn't report unless you're 30 days past due.