As
@dobre_shunka mentioned, prices have gone absolutely crazy for hotel rooms since 2020 and this year really takes the cake. As an example, here are your cheapest options (before tax) downtown for a two-night stay beginning on September 16th:
Crowne Plaza: $485 per night (never stayed, but would not recommend)
Residence Inn: $557 per night
Marriott: $584 per night
Hyatt Place: $599 per night
Tennesseean: $757 per night
The Oliver: $880 per night
In years past, those prices would have been good (fair, even) for Florida or Alabama weekend. This is Akron, though, and the rates are outrageous... yet, multiple downtown hotels (Hilton, Courtyard, Embassy Suites, Hampton Inn) are sold out this weekend and I doubt they were charging significantly less than similarly-tiered hotels.
There are no available Airbnbs downtown that weekend for cheaper than the rates listed above, which is also pretty nuts. Available Airbnbs start getting cheaper the further out you get from downtown/Neyland, but I truly don't believe that the value proposition is there to make them a good choice relative to a non-downtown hotel (rates are pretty low for Airbnbs in South Knoxville, but the area is still pretty sketchy regardless of how many brewery patios and taco shops they open).
If money is no object, go for the Hyatt Place or Tennessean. I prefer the former before even considering the price break, as I like the location on the 500 block of Gay Street for its proximity to Market Square/the Old City plus the rooftop lounge is superior to the alternative, but the Tennesseean is a little nicer overall (the Hyatt Place is very nice in its own right, though) and a closer walk to Neyland.
Otherwise, a more reasonable approach is to go with a name brand hotel in West Knoxville or North Knoxville for significant savings. There's an Aloft for $189 per night out in Turkey Creek, a Hilton Garden Inn near Cedar Bluff for $249 per night and a Hampton Inn up around Merchants in North Knoxville for $229 per night. All of these allow you to check in on Friday night, grab some dinner at one of the myriad restaurants near your hotel, grab an Uber downtown at around 1:00 p.m. or 2:00 p.m. on Saturday to enjoy Market Square/Gay Street before walking over to campus and then play the waiting game for an Uber back after the game is over. Finding a parking pass might be nice since there are admittedly few places to go in the campus/downtown area with a 13-year-old at 10:30 p.m., but you can also get an Uber pretty quickly if you don't mind paying a surcharge.