Let's see if we can narrow it down.
Turn the TV off, then put your face down by the lamp door... then turn the TV back on. You should hear (in order): cooling fans (gentle hum), color wheel (whirrrr or wheee), then the ballast (fizz, click). I think the ballast is what you're describing with the clicking noise... but let's be sure.
Do you hear all three? Fans, color wheel, ballast?
Yes I hear all of those sounds.
It's sitting in a sunroom so it's pretty much impossible to darken the room, cut I could clearly see that the lamp was not coming on at all.
Relevant? The DLP lamp market is littered with counterfeit junk. Lots of folks try to save money and shop around, then end up with 2-3 lamps only to end up blaming the TV (or the expert) when it still doesn't work. It's because they're buying cheap, factory-rejected garbage. So, friend, my question was entirely relevant. I just need to make sure you purchased a genuine, OEM lamp. If you didn't, you should've.
If you still feel my questions are irrelevant, feel free to work with another expert... but I know what I'm talking about.
Now if you'd like to continue... you say the lamp "works fine" and both your lamps "test out just fine" but I'm not sure how you determined this. How did you test them?
You can't check continuity on a DLP lamp. The filaments are open, so you'll get infinite resistance every time... whether the lamp is good or bad. It's not like an incandescent where the filament itself illuminates. DLP filaments ignite the gas inside the lamp housing... so the only real test for a lamp is to install it. If you measured zero ohms, you either measured incorrectly or there's something seriously wrong with the lamp (or the meter).
Your color wheel is fine. That's the whirrr noise you're hearing. Shortly after that, you should hear the TV trying to light the lamp (fizz, click!). It should try this a few times before giving up. This almost sounds like what you're describing, but I want to be sure. You should hear cooling fans 1st (gentle hum), then the color wheel (whirrrr), then the ballast (fizz, click!).
It's good to do this in a very dark room, so you can look inside the slots on the back of the TV for any light coming from the lamp.
The light you're seeing... is it a spark, or is the actual lamp turning on? If the light coincides with the "fizz, click" noise, it's the spark.