The other solution was to use a loop-tatted ring for the centre and this is what I tried in this version:
My centre beads are too big (but I'm away on holiday with very limited supplies!) and they pull on the centre ring because my picots were also too short to accommodate the beads. BUT, the loop-tatted ring for the centre works very well and enables both threads to be in the bead so that the beads and joins on the centre ring look all the same and it is still tatted in one pass.
Might have one more go at this…