Measure the "primary" voltage on the ballasts to be sure its at least 12VDC.Maybe there is a voltage drop across any added wiring.
All "12V" automitive electronic is supposed to be designed to run from 8..16V range (some car makers specify up to 18V as "normal range") with full performance, the HIDs are usually designed to operate from down to 6V with some performance reduction.
But you may check, whether the clicking you think is the ignitor is really the ignitor and not the converter (12V-> -400VDC) attempting to start and failing due to weak power supply (the ballasts need at least 10A capable source to operate, so reasonable size batteries use to be OK, but bench top power supplies not).
It could be, when charging the secondary capacitor, the input voltage drops below the minimum input limit (uses to be in the 3..6V range) and so the ballast shuts down. You wont register this by a multimeter, because the undervoltage is too short in time (few 100 us)...