The other way you could go is based on the other timing system we used to have, which was ran on an Apple Mac computer. We used this for multicar flying laps. The electric beam was hooked through the mouse port & a program that we ran would record those 'mouse clicks' & note the time on the computers internal clock that the click took place. you then manually put the car's number against all the clicks for that car, hit the enter & it would calculate the lap times by subtracting the first time from the second etc. The program would then take that information, record it, & print out a time slip that would be printed on a small docket printer. The info on the print out included the lap times just done, your best lap so far, the best lap time in your class, your place in the class, your place in your club, & your outright position.
We stopped using this because we could only record 6 cars over 3 flying laps & everybody wanted to do 8 cars over 4 flying laps.
Something similar could be done if somebody could write program so the mouse clicks were given the time & the calculations were done on a modern PC, with windows or such. I do know that at the last supersprint i did with the NSWRRc they had a similar timing system using stick on transponders, but i wasn't going to pry too much as timing keeping is stressful enough without people wanting to ask questions. Two start beams could be used, like the drags, with both beams assigning 'mouse clicks', then the finish line being one light that clicked twice. you'd just have to record what order they went over the line & ask who left the line first, then assign the car numbers. You could easily have two pairs or more on the track. Dead heats could be decided by duplicating the 'mouse click', then giving each the different number. We had this ability with our timing, if two cars went pasted the light together, we duplicated & neither driver was terribly disadvantaged. Maybe there's a commercial program available, probably for windows & all you do is add the necessary hardware.