Instructor pilot reported during takeoff roll at a non-towered airport they observed an aircraft taxiing across the runway in front of them; so immediately rotated and flew over the taxiing aircraft.

Narrative

My student and I were holding short Runway 35. My student made a call that we were holding short. There was a company aircraft on the downwind abeam the numbers. We then made a call that we were taking Runway 35 and that we saw Aircraft Y and would keep the downwind traffic in sight. The airport was extremely busy. It was one of the only VFR days all week. The Aircraft Y made a double click on the frequency to show that they understood. As we lined up they called that they were on base. My student was practicing a short field take-off. We did full power and held the brakes. We checked the engine gauges and released the brakes. As soon as we saw our speed was about 55 indicated we saw an Aircraft Y directly ahead crossing west to east. I took control and decided to rotate because we were too close and too fast to stop. Rotate speed in this aircraft is 45 kts. The pilot showed up at the FBO and apologized. Said they looked; made a radio call; and then crossed but didn't see us. The other aircraft in the pattern landed and we talked as well. They said that they didn't hear the aircraft make a taxi call that they were crossing. My student also did not hear a call. The pilot of the Aircraft Y apologized multiple times and knew it was their fault. You could tell they really felt bad about the situation. They said they were trying to hurry for the base traffic and didn't know we were taking the runway.