I don’t think OP understands the word gauche.

We had this party for my DD about 10 years ago. I believe we sat outside the store after the party and sang happy birthday with cupcakes.
Kids had a blast.

I have done this. It’s very easy. Ideally you’d have a second parent to order the food so it’s ready at the table when you show up the kids. Eh order a family order of chicken fingers or whatever. And you can bring your own cake to the mall, obviously no one will care.

No goody bags needed- the bear is the party favor.