We went to T20 alumni event as my son was accepted to the school and the president of the local chapter of the alumni club said that “I am sure xxx will also give you a financially competitive offer. If cost is a concern for you, talk to the Financial Aid Office. It is okay to talk to them and ask. Share your concern”.
These people are used to dealing with situation like yours. If they ask, then you can show / share the other offer because they also need evidence to make adjustments. I have no knowledge about how the approval works but I guess there is a limit for adjustment for each FAO before having to ask for higher approval with sufficient backups to support the request for adjustment.