APC transfer switch UTS10BI circuit 9 & 10 for 240V

[PerOla]

My question is around circuit 9 &10 in the switch that need to be connected to a 240V 20A circuit to provide power to the switch during regular operation. Since I do not have a 240V 20A circuit in my main panel the 

electrician will have to add such a breaker there. But since I do not have an existing 240V circuit I would lose these circuits as far as backup through the switch. 

My thought to be able to utilize these was to install a subpanel and pull a couple of 15A and/or 20A 120V circuits from the main panel there to be powered by this 240V 20A circuit. Can the legs from this 240V circuit power a few 120V circuits in the subpanel in a way so it still looks like a 240V circuit to the switch?

I understand that the 240V 20A 2P breaker in the main panel would trip the whole subpanel if it tripped but I was planning to move 2-4 120V circuits from the main panel that never pull more than 1-2 A each so do not think it ever would be a problem for the reason of overloading.

This 240V circuit in the main panel would be un-balanced but would the UTS care about that?