GC vs passport name spelling

[OskiWow 0]

I was able to get my mom a green card, which she has used to enter the US a few times already. She's coming back to the US in 3 weeks or so. While booking her ticket, I just found out that her new passport, which she just got a couple of months back, has a slightly different spelling of her first name. Both spellings are acceptable transliterations from Arabic (Tahera vs Tahra). She usually spells it Tahra. Her old passport was Tahera, which ended being on her green card (embassy had to match her passport). Looks like her new passport went back to Tahra. Incidentally, my I-130 application had Tahra but noted her "alternate spelling in passport".

Question: I think she will be able to enter with her green card (Tahera) no problem, because the CBP officer probably won't care about the passport once she's made it to a POE. However, I will have to book using the name on her passport (Tahra). Will she have problems boarding her plane if they look for proof of valid visa/residence permit at her destination (she will be boarding in Abu Dhabi or Dubai)? Would the airline staff at her point of departure insist that her Green Card name match her passport/ticket name? If so, would it help if she has her old canceled passport as proof of the alternate spelling?

Thank you