Did you check to see whether your laptop may qualify for the free battery repair? If it does, then that will also include a new keyboard. I think the coverage for the battery program is longer, but I'm not certain.
https://support.apple.com/13inch-macbookpro-battery-replacement
It wouldn't hurt to ask especially if your region had restrictions in place during this time period. You are just slightly past the deadline. Sometimes Apple will make an exception although you are over a month outside the covered period, but with extenuating circumstances regarding COVID you never know.
If you don't have any luck through normal channels, then contact Apple corporate to see if they will make a repair exception. The Apple Store manager may have more leeway in granting an exception than a local AASP, but even the Apple Store manager may have their hands tied.
https://www.apple.com/contact/
I wouldn't expect much since Apple of recent years doesn't really help their customers out like they did 10+ years ago, but it never hurts to ask. You just may get lucky. Try to be calm and polite as that does go a long way.
Good luck.