Tdy123
I’ve heard mixed things about mentioning the peer school offer. A video I watched with a panel of current college admissions officers had them all swearing that competing offers didn’t matter (who knows if that’s accurate though! 
).
If one of the main reasons my D can go to one school over another is the ultimate net cost, I’m not sure why a competing aid offer (in isolation) would matter. For some schools, a $5k award brings the school down to $23k a year - in others, a $30k award brings it to $35 a year.
If they want your kid for some reason (such that they’ve offered a moderate merit amount) I’m struggling to see how a “competing offer from a peer school” would make a difference. Instead I might say: “We can afford x amount for college. Some schools have offered aid to bring it down to that amount. She’d really like to go to your school. Can there be an adjustment?”
I might be missing something though …