I am working on a project that has a 12v solenoid valve and a 5v 3a hv transformer. I am not sure what would be the best choice here. either use 2 6v batteries one for the arduino uno and and the transformer then use a relay to produce 12v by puting the two 6v batteries in series. or use one 12v battery and a voltage regulator for the transformer.
I would go with the 12V battery used directly for the solenoid and through a regulator for the HV transformer. It should not be too hard to find a 5V regulator capable of 3A output. You could run the Arduino with 12V.