I’ve been contracting with the same part 91 department for the last 3+ years using an arrangement with JP like you suggest. I negotiated the terms with the flight department including rates, reimbursable costs, and so on. We then contacted JP and told them our terms and moved forward. I had to send all of my documentation to JP and became an employee with them, but flew contract exclusively for the 91 department. The legal department of the company I was contracting for drove the requirement for me to go through JP. I believe the big reason was to shift liability to JP rather than their company. JP had better insurance coverage since that is the business they are in.
It works out well for me, I submit my expenses and request for payment to JP and get paid ASAP. JP then bills the flight department and waits for the invoice to be paid instead of me waiting. JP’s service is expensive, they charge the flight department 50% of my daily rate for their service, but very convenient for you, the contractor.
By going through JP, it also allows JP to offer and use you for other contracting jobs they may have available.