Captain reported loss of control and autopilot difficulty during climb. After stabilizing the aircraft; The flight crew elected to continue to destination airport where it was determined that after painting; the elevator trim reference marks were incorrectly applied and the aircraft weight was inaccurate.

Narrative

Before the first flight and during accepting the aircraft I realized that the aircraft was reweighed [a few months prior]; and that the weight and balance on line; and in the aircraft folder was not the correct one.The most recent weight and balance was sent to me and I asked that App data be updated. When inserting this new CG and %MAC as well as passengers weights and fuel into the new App model; the center of gravity of the aircraft was (still is?) off and seems very far forward. This was queried with Maintenance Control and the Company Chief Pilot over several emails and calls. This remains unresolved as of today.I filed report XXXXXX - 'aircraft weight and balance does not match App on Date1.Both Pilots did separate walk arounds of the aircraft before the first flight of the rotation. I was PM (Pilot Monitoring); and Name was PF (Pilot Flying). The flight was planned for a low-level repositioning from ZZZ - ZZZ1. METAR was clear skies; 10-15 kt. winds from the SE. This had no impact on the events that follow.Preflight and start-up was normal. During the after-start checklist; I pointed out to PF that I was putting the trim 1/3 back from the max forward TO (takeoff) position; because we were light; and he confirmed the trim position verbally. Taxi and line-up was normal; and I transferred controls to PFs when we were aligned with Runway XX center line (ZZZZZ SID). Take-off thrust was set; acceleration was normal; and rotation at V1 was very smooth. Gear was selected up; FLC 160 set and we were climbing towards the first point of 520 ft. AGL; where we need to turn direct ZZZZZ1 and climb to maintain 1;500 ft. at ZZZZZ2.PF asked for the autopilot and this was engaged; and we commenced the slight right turn to ZZZZZ2. We were given a frequency change to departure; and this was acknowledged and I set it on the RMU. After this frequency change and before I checked in with Departure; the autopilot disconnected; and aircraft continued to pitch up to 15 degrees and our ROC increased.An AP FAIL illuminated on both PFD's; and PF needed both hands on the controls to maintain a pitch of between +-15 degrees; and stated he tried to disconnect the AP three times using the red AP disconnect button on the yoke; as per the required memory item. I also had my left hand on the yoke; and tried to manually trim the aircraft nose down; and the pitch increased to just below 20 degrees nose up. The manual trim wheel felt as if it was engaged (it might have been the PF trying to trim the aircraft electrically?); and I pressed and held the AP disconnect switch on the yoke. I was then able to manually trim the aircraft nose down and the pressure on the yoke immediately started reducing. We were approaching 1;500 ft. with a higher than normal rate of climb; and leveled off about 1;700 ft.; and the airspeed started increasing. PF raised the flaps as we were accelerating through 200 KIAS; and descended back to 1;500 ft.; whilst the throttles were retarded to return to 200 KIAS. We thereafter commenced the right hand turn to ZZZZZ2 at 1;500 ft. Our airspeed never reduced to under 150 KIAS; and the event only lasted several seconds.ATC asked us for our altitude; and I stated that it was 1;700 ft.; and that we had an aircraft control issue and were descending back to 1;500 ft. We were given a heading to the right and asked to climb to 7;000 ft. The aircraft was now fully trimmed; and we opted to hand-fly to the heading and climb to 7;000 ft. as instructed. We trimmed the aircraft for the climb; and I checked the aircraft trim indication; and the trim indication was very forward.I opted not to return to the busy airspace around ZZZ; ZZZ2 and ZZZ3; etc; and decided to continue to our destination for the remaining 20 minutes of the flight. I did not call Dispatch due to the busy airspace and limited cruise time so that it would give us ample time to brief the arrival and landing. The autopilot was engaged and the flight continued as normal.Upon landing; it felt was if the elevator was less effective than normal; however the landing was uneventful and we taxied to the apron.The aircraft was painted; and the rigging was never placed in the correct position; and this was not caught by Maintenance nor the other 7 pilots that flew this aircraft during the preceding weeks. A post-maintenance test flight when working on major control surfaces like the elevators would be a suggestion. [Another pilot at the company] had an almost exact experience with an aircraft that came out of painting and missed on the pre-flight that the trim was not flush in the TO position; and the Training Captain saw this and opted not to fly; and let Maintenance know. They could have released a safety bulletin and had all aircraft checked that were painted; and this would have avoided this situation. The crew could set aircraft in the take-off trim range during preflight checks; and visually confirm this with the elevator before accepting an aircraft. I suggest that this be done on all aircraft as a standard in the future. The weight and balance and %MAC of aircraft still seem wrong; and Chief Pilot's office is looking into this; however; the aircraft has been returned to service.