+1 on the lamp being most likely at fault. Here is the science primer: the lamp contains Xenon gas and metal salts. The igniter/ballast combo applies many thousands volts to strike an arc inside the lamp.
Once ignited, the arc evaporates the metal salts which "glow", emitting the bright light. A particular voltage is required to maintain the xenon/metal plasma. An old or faulty tube requires higher voltage than provided and thus the arc will extinguish.
You have very likely seen this phenomenon elsewhere: sodium or mercury-vapor street lights cycling on and off. "Dumb" ballasts (magnetics only) will cycle forever, smart (with an electronic controller) will detect cycling and stop until next power-up.
Now, regarding replacement:
there is no standard and compatibility between brands is not guaranteed. Hover, it appears to me that all these retrofit kits, sold stateside under various brands, are made by the same Chinese manufacturer.
For example, I have four CQ Lights kits on my bikes (H7 low beams and H3 fog lamps), a two-lamp VVME H11 kit in my truck and just recently I received an DDM Tuning single-H7 kit. They all have the same style connectors.
The slim-style ballasts on CQ and DMM kits appear identical and all the connectors match. This means that you have more sources for getting a replacement.
Having said that, the whole H7 kit from DDM cost me about $30 shipped by Priority Mail. Meaning, just buy the whole set - save the old ballast for emergencies. Check out:
motorcycle kit from DDM