(mainly directed towards aslp but applies to 1xx lp as well)
Broken Vessel has 525hp; stagger values are 9 hits for combo (<1s/hit) and 13 otherwise; headbanging phase starts for the first time once you deal 155 damage, so 4 shade soul hits is always enough to trigger it.
You want to always start the fight by firing 2 shade souls (where you do a small jump between the first fb and the second one), which has a few different outcomes:
-both shade souls double or one triples and the other singles (e.g. whilst bv is on the ground or dashing towards you):
For this case get 4 nail hits before the headbanging phase starts*, and then you want to get 1 nail, 1 pogo and a ddark, followed by 1 upslash, one pogo and another ddark, followed by a pogo and a ddark, finally ending with a ddark.
-only one shade soul doubles and two single hit:
For this case there's pretty much no risk of staggering before the headbanging phase starts, so try to get 7/8 nail hits in and then do your first ddark. After that you want to do a sequence of 1 upslash, 1 pogo and a ddark (all) twice, which will send the boss into a staggered state. Depending on wether you got 7 or 8 nail hits at the beginning you can either directly ddark the boss (or shade soul double if you're feeling spicy) or you'll have to do a pogo into the ddark.
The two most common cases are the ones described above, but there are some other misc starts to the fight, for example, where you get "bad rng" but the 3rd shade soul doubles (or even triples in some cases) or you try to go for three doubles. Essentially, if you get 5 shade soul hits at the start of the fight you want to follow the 1double2singles case (soul gathering a little less strict), but instead of finishing with a ddark you can get enough soul for a last shade soul.
*try not going for more than 4 because stagger is at 9 combo hits (or 3 hits if you start with 5 shade soul hits)