Unless your job requires crossing border, even if extension letter expires, you have nothing to worry about.
For I-9 verification, all you need is valid DL / state ID and unrestricted SS card. You can get both now.
I got unrestricted SS card in 2020 and updated DL in 2021. CA DMV gave me new DL expiring in 2026.
When I changed jobs last year, I told new employer I was LPR and provided DL + SS card. No GC was shown or asked in the process.
You should be able to get I-551 stamp or extension letter regardless of your N-400 as long as I-751 is pending.