My state/county requires a "unexpired Permanent Resident Card (Form I-551), including an I-94 with attached photo stamped "I-551 Temporary Proof of Lawful Permanent Resident" or a foreign passport stamped "Processed for I-551"
Wow, I didn't need the I-94 with attached photo stamped "I-551 Temporary Proof of Lawful Permanent Resident" last time.. Looks like a trip to Des Moines is inevitable, if they'll take me.
The "including" means that the types of I-551 accepted include those. But an unexpired green card is enough by itself. In any case, these are requirements for a new driver's license, not for renewal.
Regulation
761-605.25(6) says that a foreign national with "temporary lawful status" must provide documentation of lawful status at renewal
605.25(6) A licensee who is a foreign national with temporary lawful status must provide documentation of lawful status as required by 761—subrule 601.5(4) at each renewal.
But permanent residency is arguably not a "temporary" status. I can't find a definition for "temporary lawful status", but regulation
761-605.5(6)c says that a driver's license issued to a foreign national with temporary lawful status should say "limited term", and regulation
761-605.9 says that a driver's license should be issued to a foreign national with temporary lawful status only for the period of time they are authorized to be present, not to exceed 2 years, or if their status has no expiration date, for 1 year. Neither of these things correspond to your driver's license, since it doesn't say "limited term", and its expiration didn't correspond with your green card expiration or extension letter expiration. So there is some circumstantial evidence that you are probably not considered to be in "temporary lawful status" and probably don't need to present proof of lawful status to renew the driver's license (at least if you don't convert it to REAL ID). I know some other states don't require permanent residents to present proof at renewal. But ultimately this is up to the DMV as to how to implement it.
I know it says that non-citizens cannot renew online, but that doesn't necessarily mean you need to present proof of status to renew it in person. So why don't you go to a DMV to renew in person, assume that you can renew without presenting any proof of status, don't mention anything about your green card unless asked, don't convert it to REAL ID if your license is currently not REAL ID (just continue renewing a non-REAL-ID one), and see what happens? If they ask you for proof of status, point out that you got your current license as a permanent resident, which is not a "temporary" status, and your current license isn't a limited term license. Try another DMV if the first doesn't work.