I would caution against travelling out of the US in your present state - that is, to say with an incorrectly issued 10yr card and an I-751 extension letter.
The extension letter is only valid when accompanying an expired 2 year card.
By itself it is useless - for API, for boarding, for entry, for anything.
The 10 year card, when run - possibly at API/check in but definitely at the border - should show your correct status, which is that of a permanent resident with an expired 2 year card with a pending I-751...buuuuuuut you would be presenting an apparently valid & unexpired 10 year card, without a pending I-90........
You see what I am getting at here? It could open a whole can of messy worms for you to have to explain and may take a little while. Take it from me, visiting secondary every time sucks.
Could it work? Possibly. Is there massive potential for being stuck overseas and having to pay 600 bucks for a boarding foil? Absolutely. Best not to take chances.
Thankfully, there is an easy(ish) solution, which is an I-551 stamp in your passport, which negates the need for the extension letter. Get an appointment made, and when you have an appointment, take along the 10 year card, the denied I-90, and your new receipt/extension, explain the situation, and they will hook you up. Unfortunately this needs to be renewed every 12 months, but better than the alternative.