So, this changes things.
I would actually host at a site that gives you a separate room for your party and a separate menu which includes non-alcoholic beverages.
I have hosted MANY private room 3 course and 4 course sit down lunches with non-alcoholic beverages and coffee included. We were a "by the head" priced group and the restaurant gave us our own menus with their course options. While the restaurants had alcohol, it never came up b/c we had a set plan in place and everyone knew it, b/c I even sent the menus in advance with the invites, so folks could know if it worked for their dietary needs, food desires, and allergies (and everyone was out of town, so they knew they wouldn't starve on the way home
).
TL/DR - I would not go as a "regular restaurant" party, but as a catered event, and list all the details for that event. If the alcoholic is going to leave the event to go find alcohol, well, he/she might bring it in with them, too...but this way, you've limited the possibilities.
PS - I did my events this way b/c I had 2 drinkers who weren't alcoholics, but thought nothing of a few beers and then driving...and I didn't want them trying that on I-95...
PPS - Usually, I had to guarantee 15 or 20 adults, so this may not be possible for you...but if you're close to 15...