Is it permissible for a high school student to be content with merely signing in and out of school and relying on private lessons to avoid expulsion and conversion to a homeschooling student, which might deprive him of admission to some colleges and university housing?

The ruling on using communication tools in exams is subject to the regulations of the educational authority. If the regulations prohibit it, then what was done is impermissible due to the deception and cheating involved. The mere permission of teachers does not legalize it if it goes against the regulations. However, if the regulations permit it, then there is no harm in it.