2 lights on the background will be an absolute minimum here, with a large background, and all of it used, it would make much more sense to have 4 lights on the background, it will light it more evenly and reduce/remove the need for PP.
And it will have the added benefit of redundancy - if a flash head fails for any reason then, effectively, you will have a spare. You don't need fancy, expensive modifiers to light the background, you just need white reflective umbrellas, see
here
For your large group shots, again a couple of white reflective umbrellas is all that you need, there will be no advantage in having softboxes for this, it isn't in any event a situation in which you can light creatively.
For individual shots, just a softbox and an umbrella, simple.
And you won't need a lot of power either, the days of needing very powerful flash heads pretty well died with the introduction of digital cameras that can be used at higher than 100 ISO - if you ever need more power, just turn the ISO up a bit. The Lencarta ElitePro 600 has a guide number of 230 (ft), fitted with a standard reflector - this means f/23 at a distance of 10' at 100 ISO... I don't know the Elinchrom figures but they will be similar.
So, 3 x Lencarta ElitePro 1200Ws kits + 1 softbox, or the equivalent in another make, job done.