I have not used the InkScape plugin, so can’t offer feedback on that. I know I can produce what you are looking to do within LightBurn but would require a bit of work to accomplish. LightBurn does not offer a “puzzle” type tool to assist.
If I wanted to do this all within LightBurn, I would start by google searching for vector puzzle joint shapes I could use as my “Clipping” tool to separate your art into appropriate shipping sizes that can be reassembled. In the example below, I created a rectangle, then used the ‘Radius’ tool to round the bottom corners, then used snapping to align things. I had to use several duplicates of the rounded corner rectangle to complete the clipping process using one of the rectangles and one puzzle shapes, then select the ‘Weld’ tool to weld my shape (puzzle) to yours (rounded-corner shape).
I would take some time to plan this out to have a good puzzle joint pattern that would best support the assembly process, providing good strength and fit for purpose. I would also use a kerf setting in my cut to provide room for the pieces the fit back together without to much force.