You may want to do some research into Japanese wood joinery. There are some amazing techniques out there. Just one of them...
I believe this one is called the double penetration
I would make two horizontal slots along the bottom of the tray, maybe 2-3cm wide and as long as you're able to print on your printer (pro tip: rotate long skinny models 45° to get ~40% extra length by printing corner to corner). Or even better, if you've got some spare sheet metal laying around throw those in the slots.
If you're willing to change up the design aesthetic and have other materials lying around (e.g. dowels, rods, some kind of sheeting), you could also just print the "end caps" with slots for the main body.
This also has the advantage of being sturdier and maybe easier to print, but the design might be more involved.
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