Here I go again.

[EddyCurrent]

I'm only guessing here as I can't fully visualize the set up but what about friction rollers, think 'old fashioned washing mangle', the roller could be driven by a stepper motor or have an encoder fitted to measure the material position. There would need to be mechanical guides to stop the material skewing.

[Robin Hewitt]

It is tricky. If you add mechanical guides you are depending on the sheet being square sided. Rollers mean wasted margins.

Perhaps some kind of frame that you attach to the sheet end before feeding it in to the cutter?

Limiting the head movement to one narrow axis solves all kinds of problems with space, mirrors, air connects, beam catching, smoke clearance, no honeycomb required. Dreamy.