Looking for a second dog
I understand that you’re ready to add to your family. Your female Chiweenie has been processing grief too and should have an equal say in the decision of whatever dog you adopt. Besides breed compatibility, it’s just as important to consider age compatibility. Meaning dog may have a completely different energy level than a puppy and that might result in a terrible mismatch. I’d be considering what type of doggy friends she has shown the most compatibility with. If I was to guess it would be dogs that are approximately the same size and age. She should meet any potential dog in neutral space before decision to adopt is made. You want to observe relaxed happy body language between both dogs. If you see hackles raised, snarling, growling, stiff body posture,whale eye, excessive lip licking, etc. that’s not the right dog to add to your family.
There’s almost always a surplus of good healthy, well behaved dogs available for free or close to it. The shelters often waive fees to reduce the number of perfectly adoptable dogs that end-up euthanized due to lack of resources plus there’s always people trying to find their dogs a new home with their local neighbors on sites like next-door because the human’s situation changed. So I wouldn't advise you get too hung up on that this bull terrier is free. There’s lots of free dogs. Focus on the best match for all members of your family.
Breed isn't the issue here. Free is an issue to me for sure.
Bringing home another dog is another issue. Have you met the potential new housemate? Have you done a neutral greeting somewhere other than your house? I would start there.
I suggest getting a smaller dog. I have lab/ pit and had a Pom mix and the pm wanted all the attention and was not found of my bigger dog even though he would not hurt him . I think in general like dogs like smilies . Do a trail for a couple weeks if you want.