Not having seen the schematics for Rachio, nor having taken one appart, I can't say for sure if there are any downsides to doing what I am about to say, use this knowledge at your own risk:
Theoretically it should be possible to connect the individual AC input cables to the SC and SP terminals of Gen 2 Rachio (polarity does not matter). This should theoretically be analogous to plugging the power into the unit, thus Rachio should power up and work as if nothing is out of the ordinary.
In this case, considering that you've already purchased appropriate barrel plugs, this will not be needed, nor advised. But I'm putting this idea out there so that @rachiosupport could let us know why it could be a bad idea