Nice short hike to the mouth of Boquillas Canyon:
https://www.nps.gov/places/boquillas-canyon-trail.htm
Slightly longer hike along Santa Elena Canyon:
https://www.nps.gov/places/santa-elena-canyon-trail.htm
Cattail falls trail, slightly longer, but not too bad. If the falls are running (doubtful, but possible), you can wade in the water at the end of the trail:
http://www.texashiking.com/Locations/ShowLocation.aspx?LocationID=16852
It will be brutally hot in June. Go at the ass crack of dawn, and get done early. People die there every few years from the heat.
After a hike, consider crossing the river/border over to Boquillas, Mexico. Couple of bucks to get rowed across, couple more bucks to rent donkeys for the kids to ride the short but hot and dusty mile to Boquillas. If you know how to ride, the last time I was there, they had a few horses you could rent also. Kids will like the donkey ride, adults will like the margaritas and food at one of the couple of restaurants in Boquillas. Buy some tourist shit, or just walk around, shop, drink, eat. Hot, dusty but worth it. Just make sure you leave in time to get back across before the border closes. You will need passports.
https://www.nps.gov/bibe/planyourvisit/visiting-boquillas.htm
There is a nice little nature center about five miles south of ft davis on the road between alpine and fort davis. Shortish hikes, nice garden, educational crap for the kids. Probably a few degrees cooler than Big Bend Park.
https://www.cdri.org/
And for god sakes, take the kids to Balmorhea Park for swimming. Think Barton Springs in the desert with mountains in the distance. My favorite place in Texas. You will have to get reservations in advance. From Fort Davis, the drive to Balmorhea is only about 35 miles away. The road from Fort Davis to Balmorhea is one of the most scenic drives in Texas, imo. Going to Balmorhea, when you cross Wild Rose Pass, the view is orgasmic.
https://tpwd.texas.gov/state-parks/balmorhea
Good BBQ place in Marathon with adjacent micro-brewery:
https://gagehotel.com/dining/brick-vault-brewery-barbecue/
Might consider staying at the Gage hotel in Marathon. Kind of pricey but nice.
There are air BnB's all over the place in Alpine. A shitty hotel or two in Fort Davis. Fairly nice hotel in downtown Fort Davis: https://www.hotellimpia.com/
The old Indian Lodge in Fort Davis State Park is also an option.
There is also the old fort in Fort Davis if you are a history buff:
https://www.nps.gov/foda/index.htm
Finally check out the Caboose ice Cream joint in Fort Davis. Inside an actual caboose, right across from the historic fort:
https://bigbendtimes.com/2023/10/26/19-year-old-takes-charge-iconic-fort-davis-ice-cream-shop-reopens/
I don't think you'll need a gun. Too hot for bandits, and any animals are too smart to be out and about during daytime hours in June. But I think its legal to have a gun in the national parks now (better check on that). Definitely NOT legal to take it into Mexico if you go to Boquillas.