CE560 Captain reported they left their seat to check an access door warning light when First Officer was also not seated while engine running. Resulted in multiple communication and procedural failures.

Narrative

We had owners on board at ZZZ going to ZZZ1. Person A was acting as Second in Command (SIC). Before engine start; SIC came and informed the owners; we were going to ship their bags. SIC did not communicate with me before this declaration. I told the owners it's sometimes like a puzzle and I would look at it. It took me less than three minutes to rearrange and load all the luggage. Earlier in the week SIC had made many comments about owner's schedules not affecting his schedule and he had minorly delayed an owner earlier in the week at ZZZ2. I mention this to set explain my mindset on the incident day; Day 8. After engine start; we had a tail access door annunciator light illuminated. SIC was already getting up after I shut down the left engine down and right engine running; offering to check it out. I mentioned the locks as a possible cause too (Aircraft X.). SIC was walking back to the entrance; the light still illuminated. I had the parking brake set; but the right engine was still running. I got out of my seat because I felt SIC would tell the owners the aircraft was broke; which at that point it would be because of the owners' perception after a statement like that. I went to the rear of the aircraft and noticed immediately the battery door lock was unlocked. I locked it; went back to the cockpit and the tail access annunciator light was extinguished. The battery door lock had been overlooked. This was also day 8.Suggestions: Better preflight. The battery door lock was missed after connecting the battery. I should have shut down both engines and checked out the rear of the aircraft myself considering my experience that week with paired crew member. Going forward; I need to take into account the effect on my performance when operating a cockpit with personality conflicts. To further illustrate lack of communication; I was not informed the other crew member was filing an report. Like fatigue calls; this is one of those items crews usually communicate to each other.