There's nothing suspicious about getting a state ID / DL. You can get it as a LPR. I got mine a while ago on non-immigrant visa and been renewing since.
I noticed there's more questions asked by employer, TSA, anybody I present GC to. Everybody is used to seeing IDs and DLs and prefer it. Really, other than CBP / ICE / USCIS / DMV you don't need to show your GC. Maybe mortgage lender...