..meanwhile I have spent some time yesterday and today looking at this conundrum. Everywhere (hotel wise) I look at has not opened their own websites (as opposed to using booking.com) beyond 365 days out. I've tried Douglas, Ramsey, Peel, Port Erin and Port St Mary, not left many stones unturned.
But I have found this website- https://www.islandescapes.im/
As a for instance they have three properties each taking 8 people in Port Erin which all seem to be available for that week. Port Erin is a little but well equipped harbour town/seaside resort at the South west corner of the island (and the steam train terminus). I've not looked closely at the locations but honestly the 3 properties simply can't be that far apart from each other.
It is a town I know fairly well as my late uncle lived there.
Douglas is about 30 minutes drive away, none of which drive is anywhere near the TT course until the outskirts of Douglas. That could be quite a good base for you. There is also a good bus service, and the steam train normally runs TT extra commuter trains into Douglas.
It may at least be worth contacting the company and seeing what they can do for you. Even if you had to split up between Port Erin and Port St Mary it wouldn't matter as the two towns are next door to each other.
Peel would be another really good base if they could find enough rooms for you BUT part of the TT route is on your way to Douglas.