How to disable factory alarm in '97 Stang

The dealer is the only entity that is able to fix this issue. If the dealer she went to said that they can't - call another dealer. Anything involving the anti theft system needs to be handled by a dealer.

I thought you could shut off the PATS with a handheld tuner...?

Well anti-theft and Pats are actually 2 seperate modules. In the case of Mary's vehicle it sounds as if it is just an alarm issue and not a PATS issue. If that is the case it can be disarmed I believe by disconnecting the door and trunk inputs to the antitheft module so that it never knows when they are open.
You can do this right at the anit-theft module itself,it is located on the left trunk hindge brace and should have 2 connectors C403 and C404. You will have to disconnect 1 wire in each connector, in the C404 connector u will need to disconnect the T/LG wire on pin 5, this is the trunk switch and on connector C403 u will need to disconnect the DB/O wire on pin 14,this the door open signal. This should keep it from going off.
I did this on a 97 Town Car this week and it would not even arm after disconnecting these inputs.