Another ring option
https://hownot2.com/products/leash-ring?pr_prod_strat=jac&pr_rec_id=3fdc28a56&pr_rec_pid=8850950521147&pr_ref_pid=9632758464827&pr_seq=uniform
That rebraid method loses 40% of its strength, doing the math, a loop would have about 120% of the cords strength. One done with Tuck-Buries would be 178%, and certain soft shackles have a little over 200%.
The locked brummel can be done externally, instead of being inline, with the tails being...
The eye on a soft shackle isn’t a locked brummel.
Passing the tail back and forth a few time through itself doesn’t add much friction, with no locking.
Tuck- Bury has a fid and a half bury, locked brummels with the same length has a strength loss of 40%, the Tuck-Bury only has 1 1/2%.
The only way to do the second locked brummel for a loop is to unbraid and rebraid the tail.
I see Samson has updated their instructions for this splice, as well as others. The old version stopped at Step 5, I inserted the pairs through the center staggered, and just cut off.
I also see they have changed the end of the tapered tail from a fine point to leaving 3 strands.
The straight bury would be the smoothest choice, on locked brummels, the second tail would need the rebraid method, which causes a 40% decrease in strength.
This one,
https://www.samsonrope.com/docs/default-source/splice-instructions/12strand_c2_tuck_bury_eye_splice_hmpe_web.pdf?sfvrsn=c0f746b0_2
Beelines cover is also loose enough that the cover tail can be buried, rather than whipped to the side.