1. Check both bulbs. Make sure the retainer clips are on properly, and reset the bulb connectors to make sure they are tight. Those bulb connectors push down and then rotate.
2. Look at EL-39. There is a red / yellow wire coming off the head light relays for each side. I would measure the voltage on those wires when you turn the lights on and make sure you have 12 volts steady, even when the lights cut out. I think it may be easiest to test that up by the relay rather than down at the headlight, but I'm not sure. If you have 12 volts steady, then that eliminates any switch issues. I don't really think it is your switch or relays, but that would confirm the problem is downstream from the relays.
That leaves you with the stuff you mentioned. Which is all high voltage stuff. The usual test is to swap stuff between sides and see if the problem follows the part. But, that doesn't work here.
So, in order of price, I guess I would try the following:
1. Swap your old bulbs in if you still have them and see if that works first.
2. Try another brand of bulbs from a different vendor on Ebay.
2. Used OEM ballasts on Ebay have gotten very pricey. There is a 29.95 conversion kit which includes a new ballast and headlamp wiring ( both low / high beam wiring) Here is a link (seller is philipsxenon):
http://www.ebay.com/itm/D2S-D2R-D2C-HID ... dc&vxp=mtr
Those include the necessary wiring, although you may have to rewire the red / yellow power wires to the ballast. The rest is plug in. There are some other vendors you can look at as well - just search QX4 D2R ballast.
So I guess I would look at doing a kit - I don't like the idea of spending 100.00 on a used ballast only to have it fail in six months or a year.
Thats not exactly the answer you wanted I'm sure, but I tried to give you some options. Hope that helps!