General aviation pilot reported loss of aircraft control when they became distracted with a door unlocked indication. Pilot recovered aircraft and continued flight.

Narrative

Received IFR Clearance from ATC for flight from ZZZ-ZZZ1. Released for departure off Runway XX; fly heading 130; to an initial altitude of 3;000 ft. For Take Off; conditions were IFR with visibility about 2+ miles; and overcast ceiling around 1;000 feet; and solid IMC above that. Climbed out as normal and engaged Autopilot before entering IMC. Contacted Departure; provided Ident and continued climb out toward 3;000 ft. Received to climb to 5;000 and I believe a turn to a heading of 100. As we were climbing through around 3;300 ft; we experience sudden mod++ turbulence enough to cause a change in direction with strong rolling forces and up and down turbulence. Autopilot was fighting; I was gripping the yoke strongly and disconnected the Autopilot to try to dampen out any abrupt control responses. Turbulence lasted about 5 seconds or so. I tried to re-engage the Autopilot and the plane continued to turn to the left (Northeasterly). I disengaged the Autopilot and began a turn back toward my course. Had some additional short duration moderate - turbulence. While in the turn I noticed the door unlock light on the annunciator panel was lit. It took a couple of seconds for me to try to figure out how that was possible. I personally locked the door; set the handle and checked all the pins (6) for the green indicator. I looked at the pressurization gauge to see if there was any indication of a potential door leak issue but there was no change at all in the pressurization. This issue caused me to divert my attention while in the turn and the plane banked beyond my intentions and began a rapid decent in a right hand turn. I got back on the gauges stopped the turn; leveled the wings and began an initial rapid climb back to my altitude and turning back to my course. I did re-engage the Autopilot while climbing; but did not re-set the altitude. We passed through 5;000 ft. when ATC asked me if I had a problem since I had an abrupt decent & climb. I told the controller that I had an Autopilot problem and need a moment to get things set. He asked if the Autopilot was working and if I wanted to continue. I replied the Autopilot was fine and it was my mistake. I did not get into all the gyrations and turbulence discussion. Only that we are good to continue and would get to 5;000 ft. Checked annunciator panel and the door unlock was no longer lit. After that; we received further ATC headings; altitudes and handoffs to continue the flight. The rest of the flight was uneventful. I checked the door upon landing and handle was in the receiver; and all was as it should be.Reviewing what had happened on my way to my destination and after we landed; the issue started with some very strong; abrupt turbulence that caught me by surprise in an otherwise smooth and typical climb out. My decision to disengage the Autopilot to reduce input/fighting the turbulence from the Autopilot was; I believe; the correct thing to do. When I re-engaged the Autopilot; I mistakenly place the Autopilot in NAV mode (not Heading Mode); so the Autopilot started to follow the programmed flight plan for a pre-set departure waypoint (ZZZZZ); hence the reason the plane was still turning to turn left continuing to be off assigned heading. I did not realize at the time of the NAV vs Heading input discrepancy and couldn't figure out what was happening so I disengaged the Autopilot. First mistake; there was no real reason to try to re-engage the Autopilot. Should have just stayed on the Flight Director and fly back to the right course/altitude after the turbulence and advised ATC of the turbulence. In short; simply fly the plane; then sort it out. This just added to another distraction of heading. When I started to turn right to get back on course; I noticed the door unlock light on the warning panel (never saw this illuminated before). This became an additional distraction by trying to analyze if it was an issue. This should not have been a real priority at that time. Looking back; should have only noticed door unlock light and gone back to the Flight Director; fly the plane and waited until I was leveled and back on course for further investigation. Since we were at less than 5;000 ft. there was no urgent need for pressurization at that moment so I should have focused on the other tasks at hand to correct the deviations. I did not experience any spatial disorientation at all during these deviations.The abrupt turbulence began a string of actions on my part that; in reality; aggravated the situation and only further induced altitude & heading deviation until corrected. I have all my currencies for instrument/approaches and day & night landings.