I never intentionally made a through mortice on a buttjoint, but is it necessary to plunge through both pieces at the same time?
If you do it like a regular buttjoint, but adjust the cutting depth so you plunge through the board (probably best with a sacrificial piece below it to prevent tearout) this way you avoid your corner clamping problem.
/edit you could make a simple clamping jig, for example two square pieces of board with a few holes drilled in for using screwclamps that are out of the way, maybe in combination with double sided tape for the buttjoint itself.
I tried making a quick visual aid in paint, the green bars are the screwclamps.
/edit 2. If the pieces are just too small, and that's the reason the screwclamps don't work, I would just use double sided tape or if you only need to join one side Cheese's method.