Air carrier Captain reported the door handle moved during a maintenance verification flight to check the handle and cabin did not fully pressurize.

Narrative

ZZZ maintenance verification flight. Aircraft X.Door 2L issue. Cronic history going back almost a month. Maintenance onboard to observe door during takeoff and climb. Load planning notified and final weight & balance correct; showed 3 SOB.At some point during the climb; the door handle DID move on it's own; approximately ½ inch but no abnormal cockpit indications. Upon reaching cruise altitude and consulting with onboard tech; he asked if we can land and takeoff again to see if it was a repeatable occurrence.During that time; we were about equidistant between ZZZ and ZZZ1. I consulted with Dispatch; Sector XX; by voice. I suggested that we can land ZZZ1 and depart again; avoiding congested airspace and landing fees as well; especially considering another likely takeoff and landing cycle.Dispatch was agreeable and we coordinated with ATC. Landed ZZZ1 uneventfully. Further consultation with onboard Maintenance trying to verify what the door was doing; or not doing properly.Direct contact with Dispatch over several calls; new FP (Flight Plan) by email; signed through ACARS; new release; final weight manifest. All boxes checked. Departed ZZZ1 for ZZZ. No abnormal cockpit indications but onboard Maintenance had concerns about and was measuring handle movement. Landed ZZZ uneventfully. Everything was coordinated through Dispatch but apparently there was a disconnect between us and Operations. They thought the divert was due to an emergency. It was not. Maintenance just wanted another complete takeoff and pressurization cycle to observe the door. It was discussed with and verbally approved through Dispatch.There was also some confusion of whether Maintenance was supposed to be onboard for this flight at all. Load Planning was aware and our Final Weight Manifest showed proper SOB (3). Confusion came on the ground in ZZZ1. Names and file numbers were given and all seemed normal in that regard. It seems that some people were in the loop and others were not; thus creating confusion between departments.In hindsight; the only 'event' that could be considered questionable was my failure to check the AAL (Airport Authorization List). I did not realize that ZZZ1 is an approved alternate airport but not a regular airport. It was not listed as an alternate on our Flight Plan but we made it our new destination when we diverted to there with Dispatch approval. I believe that since we did not originally or intentionally plan to land there; that it's use as an alternate was justified.Return flight to ZZZ uneventful. Only got to max of 7.5 PSI cabin pressure on return flight. Maintenance still had concerns with the door handle moving. Last I saw; plane still on ground in ZZZ.