If everyone is rocking mostly 210 gear with 205 weapons AND you're stacking all your buffs for a full opener on the HoP hand, I have no idea how else to say this:
Your group is honestly just not very good.
The only reason you should be failing the Hand of Pain dps check at this point is suboptimal play or having two dps get stuck with Damage Down in that phase. My group has cleared the check easily with the first Damage Down not getting passed. We've cleared it with someone (me ;-; ) dying in phase 1 and having Weakness through the check. Someone isn't optimizing things.
Outside of that, it could be party comp. If you have SMN, SCH, BRD all multi-dotting the hands, you're losing a LOT there, requiring the melee (and MNK should be using Rockbreaker, so he isn't doing max ST either) to do a hell of a lot of extra damage to kill the check in time. Someone might also be attacking the wrong one. Are BOTH tanks hitting the right one? Is either tank in tank stance there? (Neither one should be; you should have plenty of CDs you can burn that you don't need for Drainage)
Basically, ranged DPS who get gains from multi-dotting/aoe should be multi-dotting/doing aoe. Tanks should drop tank stance and both attack the HoP target. SCH should be 100% dps uptime there and multi-dotting with all Broils focused on the main hand. If you're close to failing, it's worth it to have BRD/MCH not multi-dot and SMN stop using Painflare and use Fester instead, but there's no reason at all that people should just outright stop ALL AoE to clear it.
tl;dr: It's not your fault. You shouldn't have to hold CDs for it. Parse. Tell your group to get good. Figure out if someone is hitting the wrong hand. ONE COMBO on the wrong hand can cause a wipe, especially with AoE going on. I know that from experience. </3
The second Wash for me is usually ID>Dis>CT>4th>Phleb>ID on main hand through wash (following him the whole way, usually with Elusive Jump used) and then tab-target to the other to drop Dis>CT>[life surge]>4th and there's IR somewhere idk, but after that is HT>TTT4 on whichever hand, followed by ID> [hop target] >Dis>CT. And keep the gsks flowing. You shouldn't be holding buffs for the HoP. If you have a NIN, TA is pretty baller for killing the check. If not, you're gonna need SOMEBODY to take a hit and stop multi-dotting, more than likely, if you wanna clear it consistently.