What can I use in my own kitchen to fashion a panini press?
Assuming that you don't mind the bottom of your smaller pan getting dirty, you could put the sandwich in a heated larger pan and place the pre-heated smaller pan on top.  According to your question, you've got this this in own mental experiment.

The next step would just require you to borrow from mechanical engineering and some cheap root vegetables.

Mechanical designers use a widget called a standoff.  It just help create a gap between two surfaces.



Couldn't you use the same concept with carrots?  You could just surround the sandwich with equal length (maybe 1/4-1/2" shorter than the sandwich) cut carrot "columns" for a carrot "tripod" of sorts.  Bread will certainly cook faster than carrots and the used chunks could be edge-trimmed if they got blackened and consumed.