I got Tom in my campsite and invited him to live in my town. I tried TT-ing to plot reset him but no matter how far I go in the future or past he is still in his igloo! I am very worried...
Campers usually move in two or three days after they've accepted the invitation. Are you sure you saved after he agreed to move in? And has enough time passed?
If you still have Tom as a camper go and talk to him and see what he says - he should say he is moving in.
You can talk to him with all your characters and reinforce this (if you want, but not necessary).
Then I would do as advised by Plupap. I would play normally and open up each day following Tom's agreement with a new save file if you want to manage where he locates his house.
With the TT have you saved/used the Mayor to load? Are you sure it is still Tom in the igloo?