In the Chattanooga area, we have stayed at the Shallowford Rd. exit. I think that is Exit 5. But it has been a while and we did not need a pet-friendly.
There are a lot of hotels at that exit so if you check Trip Advisor or something like that and narrow your search to that area of I-75, you should be able to find a place there for you and yours.
PS: I think part of that area is called Hamilton Place. (It probably has been about 5 years since we were there but Trip Advisor should have recent comments.)