You need to apply for GE using the name as per the GC. If you use her maiden name as per the passport it won't work. I went through the same. Had to go back, change the name of the account and then continue adding documents. You will have to input both the GC and the passport data anyway as well as then fill out the line that asks for any other names she's known as (her maiden name, in this case).
No issues getting approved at all. Officer didn't even mention it at the interview (or ever during any border crossing to be honest). I'm finally just getting around to replacing my passport as it's expiring this year and will update my name in the process. Downside is you won't get PreCheck for those international flights since they have to be booked with the passport name, but apart from that it's not an issue at all.