Your CFI can sign you off to fly back and forth between your airport
and his. You only need a solo sign off and a sign off for the repeated.
You DO NOT need cross country training because of the close proximity.
I'm not sure what your CFI is concerned with.

I teach out of a non-tower airport and always sign my soloed students
off for solo flight to a local controlled field 10 nm North. That way
they can practice going somewhere close and talking to the tower etc.
There is no reason for a solo'd student to not be able to make such
short trips by themselves.
-Robert, CFI