A Tower Controller reported an aircraft that was instructed to hold short of a taxiway taxied across the runway at the same time as an air carrier aircraft was taking off.

Narrative

Aircraft X had been taxied out of FBO via taxiway 1; taxiway 2 hold short rwy XX with a good read back. While they were still south on 1 I coordinated a crossing of runway XX with local and taxied them 'Cross rwy XX at taxiway 3 hold short taxiway 4 ' with a good read back. 3 is the last taxiway before rwy XYL. Aircraft X never made the turn on taxiway 3 and taxied straight across rwy XYL while Aircraft Y was rotating. I yelled 'stop stop!'; Aircraft X replied something along the lines of 'we got it; we're crossing' or something similar. Aircraft Y overflew Aircraft X at low altitude while they were mid runway. I then instructed them to hold short of runway XX at taxiway.I was not taken off position after the incident to fill out a report or review the information because of inadequate staffing. I had to work the rest of the busiest departure bank of the evening shift for another hour before I got a break. Then again I only took a relatively short break before going back to work. I didn't review any of the information; only talked to the OM who told me that Aircraft X had read everything back correctly.