I love all that you are doing to support your child.
The companionship of a dog can be wonderful. I have a dog that I apply Revolution to once per month in the warmer months. I don't love being around the toxins or applying them to her, but I cannot risk the flea and tick potential otherwise. I know there are more natural treatments, but I have not found any that seem to be effective enough for my level of New England tick paranoia. I still don't let her onto the furniture, though. Maybe the dog could hang out on top of a white sheet you wash each day, and your child could get in the habit of doing a tick check? Also, when you take the dog outside, you can stick to pavement, very short grass, wood chips, etc. to minimize tick habitat exposure. But that's tough because every dog needs the opportunity to run sometimes. Maybe a small enough dog could just zip around the house.
I strongly encourage you to get a low-energy breed that is also a low-energy individual. A dog that requires a lot of exercise each day probably won't love to hang out with your child and will take up so much of your time. Get an adult dog from a rescue, as they will have been evaluated for temperament, will have less energy than a puppy, and might have some basic training already (house training, sit, etc.). A rescue also should be able to tell you how often that particular dog needs to go out. (I had a Yorkie who needed to go out only three times per day, but most small breeds need to go out much more often.) Training is essential, even with a small dog, for their own peace of mind. Training a dog can feel like a full-time job for the first couple of months, and everyone in the household needs to be on board (e.g. no one can pet the dog while it's jumping up on them or the training other people are doing about
jumping will be undone). A rescue should be able to tell you about
the responsible training classes in your area. Petfinder.com can be a good place to start, as most profiles include lots of details about
the dogs involved.
It is possible with a tiny breed to keep them entirely indoors, but I have heard some controversy about
whether that is good for a dog psychologically.
Getting a dog is not a dumb idea, but in my experience it might be a labor-intensive idea.
If you are feeling overwhelmed at all, then I'd have to advise against it. My family has a dog and a cat, and while they are the best dog and cat, my partner is weary of them. Before I got sick, there were two of us to pay attention to them, tend to their constant needs, etc. Since I got sick, I know my partner mostly views them as a burden, and I feel really guilty for not being able to ensure the sort of life for them that I know they should have. They get good nutrition, medical care, walks, comfy spots in the house, but not much else. We regard getting a pet of any kind as a genuine lifetime commitment to the wellbeing of that creature, including learning about
them and what kind of life makes them happy, not just what we want from them. But that's much harder to ensure when I'm not functioning at (or near!) 100% for four years now!
Do you want a dog? The person who will be doing the walks in the cold or rain, morning and night, should want a dog. The person who will be cleaning up pee and vomit accidents, doing all the laundry, paying the vet bills and grooming, feeding, and scrubbing slimy water bowls should want a dog.
I loooove dogs, but they are super gross and a lot of work. Darling creatures who are way better than humans, and are all heart, but a lot of work.
Is she willing to consider a different type of pet? Rats are extremely smart and interactive. They will come when called, play fetch, play hide and seek, etc. and are tiny and easy to care for. Plus no ticks.
As a prey animal, they do require some privacy and quiet time, but they are more likely to want to interact with humans than other rodents are (rabbits, hamsters, etc.).
The right cat also can be just as cuddly and fun as a dog, but once again I'd recommend an adult cat from a rescue so that you can know more about
its personality before committing.
Whatever you decide, good luck! I am a big fan of pets. I hope you find the perfect fit.