Thanks for all the responses everyone!👍🏻

As usual I think I was trying to over engineer the problem and make it so that an elephant could walk on it but it seems general consensus is that a simple baton running under the stud wall but overhanging the gap with some screws on both sides of the join into the baton should suffice.

I’ll probably still put a couple of triangular wedges in anyway for my own peace of mind as I’m a bit ocd about jobs being solid, but very much appreciated, thank you! 🙏🏻 🍻