The only thing they asked: "Your car has US plates?". I said, "It's a rental". She said, "Okay". Nothing more.
I provided them with both: a
copy of the COPR (I had my original as well, safe inside, but did not provide that) and my passport. She just glanced at the COPR, scanned my passport and we were on our way
My most recent cross-border drive was when I moved to Montreal in August 2018. No PR card. We flew into Burlington, VT. Picked up an Avis at the airport, drove to Montreal (passport+COPR only), dropped off the car in Montreal. Simple.
Rental cars for personal use are NOT considered commercial and the CBSA gets that. Your plan of driving across will not have any problems using passport+COPR. Drive safe!