I have to wonder if some kind of fly in fly out arrangement like is used for a lot of other jobs in undesirable locations would work. I've got a friend who's a nurse/paramedic in an indigenous community in the australian outback, and he works 6 weeks in the community with a per diem and company house, and then has 6 weeks to travel or live in Brisbane between stints. A school could do quarterly swaps of teachers, so there's a Q1/Q3 teacher and a Q2/Q4 teacher each working 9 weeks at a time.

$100K would certainly garner tons of prospects. Rural schools around me only pay $30K