If you are like me, using expensive refractory to cast slabs is intimidating, but spending an equal or greater amount of time and
energy to find alternatives seems like fun!
In that spirit I offer two ideas:
First, binding the outside edges with wire.
Nichrome wire can take heat and is easily ordered.
A shallow cut down the center of each 2.5" side
should be
enough to hold that wire in place once it has been tightened.
Second, "peg" joinery, using ceramic pins/nails.
They are roughly 3/4 " long and should fit into a 1/8" hole.
Made to survive the direct blast of a kiln, they should do fine sandwiched inside two bricks
Just some ideas.
https://m.media-amazon.com/images/I/31pFX3heW5L._AC_SL1500_.jpg