I'm surprised no one has come back to you on this yet; hope they do - I'm interested too.
Sorry, I can add much value myself.
I don't tend to sail much in the shallow water where I am, due to both the random rocks and (in the known soft/sandy areas) the scattered lumps of scrap iron/concrete etc that act as mooring weights; I fret for my some-what fragile board, and so I generally motor when needing to get up wind in very shallow water.
I didn't fret quite so much for the Lugger (with its steel "board"), but then I rarely was in a position to sail upwind in the river anyway - the boat simply wouldn't point to any worthwhile extent.
If pushed, in the BRe I try to trim out/reduce any weather helm by adjusting the mizzen sheets for balance, and on a few odd occasions this has worked for me. That said, I then always have at least some board down, rudder too (of course).
Regardless of the board, trimming the rudder up significantly will cause a dramatic load-up of the tiller, independently of board depth.