Seat Design Finalised
In Australia, our equivalent to the USA's Home Depot is Bunnings Hardware. Tradies use them, handyfolk use them, they carry nearly every imaginable item. One item I've used a LOT of is the Carinya Make-A-Bracket line.
About 7 or 8 pieces of the 200x200 (shown) should be enough to make the metal tray for the play mat foam padding. Just add some heavy duty grommet strip to protect flesh from the metal edges and you have a light, strong seat.