The key to Louhi is the amount of max dps you can accomplish in that first minute. Mana is a total non issue for the fight if you have 2 necros with blighted ones and void feated, 2 rangers with exhausting traps, and DT's cycling thru mystical banes. If you have ToS with Coils and BS's draining mana as well, even easier.
If you can get Louhi down to 25-30% in that first minute, which we do in all our alt raids, the fight will only take 4-4:30. If you have gimp dps and can only get 50-60% (which I've unfortunately seen in others Louhi fights), the fight will drag on forever. And the longer the fight, the more useful mana burns will be, so yes, if your raid has poor dps, then you will need all kinds of extra mana burners, including sins probably.
The number one issue is to minimize the amount of time you spend after 4:00 into the fight when you have to deal with the stacking debuff, and to negate that, you must work on maximizing dps for that initial burn in the first minute before the 90% mitigation shield goes up.
Typically if mana is an issue, that means you are taking too long killing the boss, which goes back to what I stated earlier about the initial 1 minute burn being crucial to the fight. Spending more time in the post 4:00 mark after stacks go active will make it even more difficult to manage mana as your melee mana burners possibly stack themselves to death.
As for the tanks comment, we have 2 guards tank Louhi (with varying amounts of protection gear, depends on how long they been in the alt raid, we cycle them out). Our DT's are pet spec'd and they space out their dreads to give some relief to the 2 guards, but they are in frenzy stance for the entire fight (unless they need to swap to defensive to help tank). Conqs are in full dps carnage builds.
TLDR - Initial 1 min burn is crucial in the Louhi fight and mana drain for sins is definitely not needed.