Yeah, 115 Falcorr > Hydra. With Hunaphu you'll be able to more feasibly work your way up to the 5th floor for gear. On the 5th floor you might need to do some "creative sac'ing" or kiting depending on what you have unlocked already, as the Archaic machines are considerably tougher, and are more difficult to straight kill as a BST with only ~200 DEF(no gear), weapons and accessories. There's always the possibility when you first enter ZR that the first pole gives you one of head/body/hands/legs/feet, in which case you'll have a significantly easier time killing mobs than you would by having just an iLvl weapon.
As a testimony to your idea of going as BST x 3 to this run, I've had success with:
BST/DNC, BST/SCH, BST/SCH
--/SCH with Arts up for landing sleep, Dark Arts if you want to Manifestation it for a bad pull; helpful for the climb since pulling things 1 by 1 isn't always ideal. Accession Blink/Stoneskin/Regen II are really helpful for sustainment on floor 5+, by which time everyone should have JA/Subjob to optimize use of these spells.
--/DNC should always be tanking, keeping steps up, and popping a VFlourish as needed(i.e. on Restoral), or DFlourish+Charm if you need to kite.
You would be better off just bringing an actual support class and just 1-2 BST if using BST to do the killing, but the above works if your only options are BSTs.
For the odd KClub BST (or OAX)that's curious, this is much easier to manage as you'll always have TP for cures, and wont really have to worry about sleeping/kiting anywhere near as much if at all
Once you get past the first couple of Archaic Gear(s) on the 5th floor you'll be rolling along smoothly. Killing the gears without gear or pets is without doubt the hardest part of the climb if you're running with only BST. Hope this was helpful.