>>2931047
If I understand you correctly then I would suggest that you build the entire piece out of oak, with the solid lip you depicted only on the front, and then simply take some MDF and screw it on so that it can move. You could hide it entirely by screwing from the top before tiling.
If you'll excuse the DaveCAD I'll try to explain how you'd hide all the shenanigans. You drill 2 (or more) sets of countersunk holes in the MDF with the front being the exact diameter of the screw so it can't move, and the back being oblong so the screw can move as the oak shrinks and expands underneath the MDF. That way the tiled MDF plate says aligned with the front. Same things for the sides, of course, since you're tiling all the way around.
I wouldn't bother with the expansion gap on the top then. Just join the oak top directly to the oak sides with dowels or whatever you were going to and it'll be solid. It will not move meaningfully in that direction.
It's gonna be egregiously expensive and I don't know about your assertion that you won't have to worry about cutting square. You're definitely going to want to plane all the inaccuracies off once you're done assembling it.