I would think that you would do the ring first, mobius brummel (in the riggers apprentice) looks like what everyone is using. I would do some tight siezing by the ring, i don't know why that fell out of style as a way to snug up to a ring or thimble. For the double head i thought that they either did a friction hitch because it can't slip over the other ring, or another piece of tenex, same size, splice ring on, cut shorter, placed the core of the long one inside the shorter one (2nd ring), and then bury the tail of the 2nd ring inside the first. That would self equalise with use, and look very clean. I'm pretty sure Richard Mumford did a friction hitch using a hollow braid, that man is a genius with rope.
It could be done like this:
For your eye end, either a locking brummel or straight bury but keep the bury the entire length of the sling until it branches out for the ring splices. Then you just do two straight burys, one for each ring, lockstitch, and Bob's yer uncle.
long-tailed locked Brummel around first ring, then locked Brummel around second ring, then locked Brummel/bury to finish that end. For an eye on the other end, do an inverted locked Brummel. Brion Toss shows this on his splicing video. You pull the end all the way through, til it's inverted, then pull the second through to inversion, then undo them to create the locked Brummel.
EDIT: I'm thinking the rings themselves are NOT in Locked Brummels, as there is nowhere for them to go with the Locked Brummel securing the base of the split. I did clinch the rings in tight, then stitched each bury so there's no way it can shift and lose a ring.