How about setting up a dry lease where Company A leases the airplane to Company B for whatever the agreed rate. Now company B is operating the flight, so it is a P 91 operation. Company B supplies the pilots and gets added as a named insured to the policy.

Just make sure that Company B is free to use whatever pilots they want. If Company A sets as a condition that Company B uses Company A pilots, then that is really a wet lease and becomes an illegal operation under part 91.

http://www.omnijet.com/topics/drylease.pdf