I know you probably want to make your daughter happy but if a BBQ is what you can afford then she should just deal with it.
Also, I can think of lots of free places to throw a party but they are all outside!
The botanic gardens and the water gardens in downtown fort worth are really pretty and cool places to hang out, but you're right it will very hot this weekend.
But if you're willing to spend some money there is a restaraunt called Sprouts in arlington off Collins that is really young/hip, has an upbeat atmosphere and they serve vietamese food. They also have those wierd boba drinks. It could be exciting trying the different drinks and desserts if you're not used to that sort of thing.