That’s pretty much exactly what I did when I was test cutting that thick foamcore…I just taped a bunch of 1/4 inch ply offcuts together and stuck them at either side of the bed, and taped the foamcore between them. (Very scary looking setup actually, but it worked just fine.)
So you can make your side setup tall enough that the top of the cardboard is slightly below the object to be engraved (about 1/4" lower) then just reset the thickness for the taller object to be engraved and drop the boxes down into the hole cut in the cardboard. Make sure you put a piece of scrap wood under the cardboard the first time you cut it, so you don’t fire the laser into the metal underneath, but then you can remove it for the engraving part. And just keep reusing the hole - you only have to cut the jig once, then you can engrave as many boxes in a session as you want.
Once you close the app, you will lose the relative placement for this method, so that means either cutting another piece of cardboard the next time, or if you want to try to hit specific repeatable positioning after closing out a file, @takitus has a specific tutorial for it in the Matrix.
Either one of those should work. (It does add an element of challenge when you take it over tray cutting levels, but it can be done.)
Later Update: We have a report now that the full height of 0.50 inches is now obtainable! Yaaaaay!