Ashbow,
This seems fairly unambiguous, and you should be able to conduct the task as described.
You and the other person on board are both employees of the company who has commissioned the work, and are doing the task as part of your duties for that company. There is no third party customer or operator, thus it is a private flight. No AOC is required.
You are being paid by your employer to perform duties which include flying an aircraft. Thus, you are being paid to fly, and can only do that if you have a valid CPL; you imply that you have one, so you are fine on that front.
Your company is renting the aircraft from someone else. The company will presumably want to rely on the insurance policy of the Flying School. This is the only area where you need to be careful, and you will want to assure yourself that the mission is allowed by their insurance, and that renters, and their employees, are automatically considered as additional assureds under the policy. This is not always completely clear cut in the policy, so you would be wise to check before you launch.