Small aircraft on departure roll; observed aircraft crossing runway down field; elected to maximize climb and over fly incursion traffic; causing a pilot deviation.

Narrative

I landed at ZZZ Runway XX exiting at Taxiway D. I asked for taxi back to Runway XY. I was given instructions by Ground to proceed to Runway XY via cross Taxiway A; left on Taxiway K to Taxiway G. At Taxiway G I did my run-up. While doing my run-up; I monitored Ground and heard exchanges from aircraft and Ground that were taxiing Taxiway A for Runway XX. I changed from Ground frequency to Tower and told Tower I was at Taxiway G ready for takeoff on Runway XY. I was told to hold short. I was then given permission to take off Runway XY. I proceeded onto the runway and applied takeoff power. As I was building speed; I observed a twin engine aircraft traveling from my right to left on Taxiway A approaching Runway XY. The twin entered Runway XY. I determined that I did not have enough runway to stop and started an emergency climb. I crossed over the twin while he was on Runway XY. I called the Tower and asked if the twin was supposed to stop at Runway XY. The Tower affirmed that the twin was told to hold short Runway XY. I was asked for a phone number from the Tower. I supplied a number and proceeded to my destination. The problem arose by the pilot of the twin not holding short of the runway.