Courtyard by Marriott Fort Worth University DriveWalkable to:
Woodshed (1 minute walk, decent beer, dog friendly patio, pricey)
Rogers Roundhouse (5 minute walk, Great dog friendly patio, good menu that fits kids, not too pricey, good beer selection)
HG Supply (Walk down Trinity trail to the east under University bridge for 5 minutes, good food, hipster-ish vibe with weird menu, but great dog friendly patio)
Downtown HotelsGives you more options for food, but not for dog patio type stuff.
Residence Inn by Marriott Fort Worth Cultural DistrictWould be a long walk to most anything along West 7th which is a busy street, it would take you 15 minutes to reach Rodeo Goat or American Gardens. West 7th turns into a bar hopping atmosphere after 7 or 8 but most places have a good atmosphere during the day.
Or you're next to the park where you could walk south to the restaurants listed on University but it would take you like 45 minutes to walk to the Woodshed
Stockyard HotelsCan't think of any dog friendly patio places in the stockyards, but cool to walk around for an afternoon.
My suggestion would be the Marriott on University Drive and walking to Rogers Roundhouse. And that hotel will be easy access to I-30 or University which will take you wherever you are looking to go. TCU is a 5 minute drive south on University, West 7th/Museum District is a 5 minute drive North on University, Magnolia is about 10 minutes, and so is Downtown.
Here is Rogers Roundhouse website
https://www.rogersroundhouse.com/If you're looking for a brewery, you could drive to Wild Acre which would take 10 minutes or so. I don't believe they have food, but a good brewery with some yard games for the kids and a dog friendly patio.