In that case I'd stick to an Amazon wishlist, a gift card or cash.
POD. For teens an Amazon gift card ought to be pretty useful.
I guess my larger thought is, the question is not really, "When is the right time to ask for birthday gifts?" Because with your SIL, there is
no right time. There's no magic timing or phrase that's going to make her suddenly change her attitude.
So, I would probably ask a nominal question about a month before, just so she can't say I didn't; and then do whatever was easiest and least time-consuming for me, while still being a decent gift.
I consider a gift card a decent gift, or a check would be fine too. If the OP doesn't feel those are decent gifts, then something from a wishlist (as long as it isn't too old) should be fine. Or, a lot of online stores will divide their goods into categories, so you can see what's generally popular among teen girls, for example, and get something from that. As long as it's not difficult to return, and not totally out of the ballpark--like baby toys for a teenager, for example. I find that popular books are often a good choice.