Is there any nice way of identifying which model of Peerless Bathroom Faucet I have?
Since the cartridge is out, take it to a plumbing supply house not your local big-box home improvement center. 

Odds are very good that they'll either 


know off the top of their heads, 
be able to find a part number that you're missing, 
or just simply walk to the back and open 12 boxes looking for one that matches and sell you that one.


One additional tip: Call ahead. Some places will only sell wholesale, i.e. to commercial plumbers. Others will sell retail, i.e. to you.