Captain reported an aircraft began to taxi across the runway during takeoff roll. The Captain elected to continue the takeoff and climbed over the other aircraft as it crossed the runway.

Narrative

Runway incursion by another aircraft. We were cleared to takeoff [runway] XX at ZZZ on Aircraft X. I was the PM PIC in the left seat. After making the 70kt call; I looked up and saw Aircraft Y at my 11 o'clock position taxing southwest towards runway XX. I watched them for a second to make sure they were not crossing our runway. They appeared to be near the very far end of XX. I looked back at the instruments and called V1/rotate. I quickly looked back up to see that Aircraft Y was not stopping and was crossing XX. As we got airborne I saw they were taxing southwest on taxiway 1 and crossing runway XX underneath us. I reported the incident to tower after liftoff; and was told that they were looking into it. After departing the ZZZ terminal area I started recounting the events. When I first saw Aircraft Y moving; they were not a threat yet. It wasn't until just before reaching V1 that I began to get concerned. I quickly determined that if they did not stop; our ability to overfly them would not be in question. They appeared to be near the far end of the runway. I also wasn't perfectly certain of the taxi layout in that location and thought that possibly they were on a taxiway that was beyond the end of XX. If we made a delayed and late rejected takeoff on the wet runway; we could possibly make the situation worse. It wasn't until after reaching V1 that it was evident that Aircraft Y was going to cross our runway. At that point the decision to continue the takeoff was committed to and we easily overflew Aircraft Y.