Installing 3-phase power would be silly overkill in this instance.
cvanc simply needs to double and triple check that everything really is 120 volt and then supply it from both sides of the available split 240.
As suggested multiple times above, this can be as simple as wiring two of the phases to one side, provided that the imbalance doesnt't cause an overloaded neutral, though that is unlikely as long as it is on both sides of the 240, not just all on a single 120 (which would be silly anyway.)
If the imbalance will be huge or the OP wants to run from a smaller breaker, I really can't see why it would be too difficult to move even just a couple of the supplies from one phase to the other to balance things out, even if he doesn't want to move everything from the third phase.
I still think it would be pretty easy to move
everything from the blue phase (or whatever one was easiest) over to the other two, but surely it can't be too difficult to just strap two of them together at the supply and move just a couple of the supplies from the now more heavily loaded "double side" over to the other phase.
I can't see why that would be very difficult.
Of course, I haven't seen any photos of the actual unit's internal horror show either, though.