They
stated they did not do any training, BFRs, or IPCs in a non-school
aircraft as their insurance would not cover them.

What insurance? Do they have professional liability coverage for their
instructors? I highly doubt it. More likely they just want to rent
their planes. However, the insurance issue is not absolutely
impossible.

Since I am the PIC and it is my airplane is it not my insurance that
covers the flight?

Your insurance covers you. It is conceivable (though highly unlikely)
that the insurance company would choose to subrogate against the CFI -
for example, in the event of a midair (since you were presumably under
the hood and the CFI was responsible for collision avoidance). Of
course there's probably nothing to get from the CFI, but if he is
provided by the flight school (either as employee or contractor) and
they take a cut of the money, they also bear responsibility. It's not
likely, but it's not impossible either (especially in the event of a
midair).

The real solution? Find an independent. Odds are you will get better
training anyway.

Michael