Firstly, to everyone, thank you for taking the time to respond and for giving me the opportunity to learn from you all.
What LEDs? (From Amazon) Cutequeen White 42mm(1.72") 8-SMD 12V Festoon Dome Light LED Bulbs. Unfortunately, I don't have any technical specifications at this point. I do know there is zero (measurable) resistance across the bulbs vs. 1.5 ohms in the OEM 212-2 incandescent bulbs.
Also note, I used these same LEDs for the exact same application in a similar but older 1996 vehicle, and they work without exception...wonderfully. It's this newer (2004) vehicle's electronics that seems to monitor and/or regulate things (now) that's added much more complexity to an otherwise simple task. I confirmed this with a mechanic I spoke with on Friday who stated (from schematics) there is a "module" that also serves the instrument cluster as well as these lights. It uses "drivers" to supply the requisite 12VDC when switched "on" and there is resistance vis a vis the incandescent bulb (1.5 ohms).
As for the measurement phase, as best I understand it, it's constant while there is no resistance. In fact, I think this is precisely why the LED lights glow faintly while the ignition is on. Oddly enough, once you open the glove box, which has a tiny, single incandescent bulb, then the map lights will work independently on/off normally just like with the incandescent. However, using LEDs in both the map lights and the dome light--while all in the off position--and with the glove box closed, all three lights faintly glow due to the constant 4.55 volts reaching them. So while this isn't enough to light them, it is enough to make them glow. So the measurement phase seems indefinite while there is no resistance like with the LEDs installed or the quick connect plug is disconnected (that serves the map lights).
For what it's worth, here are some other things I've observed: If I put a LED bulb in the dome light but leave the two incandescents in the map lights, everything seems to work like normal. However, if I put LEDs in the the two map lights and an incandescent in the dome, only the dome light works when I turn on all interior lights via the dimmer control [position] switch yet the dome and two map lights will all light when a door is opened. But with doors closed, if the glove box door is opened (which automatically engages the tiny light inside the glove box), both map lights will work like normal with LEDs installed; however, if you swap the dome light to an LED as well, it will glow faintly in this situation.
So, prior to reading your input and responses to my question, I [guessed at] soldering in two 100 ohm 10-watt wire-wound Cermet resistors (one resistor soldered across both map light bulb receptacle contacts...no resistor for the dome light) . While these did get pretty warm over a constant 5+ minutes while left "on" for testig, they worked perfectly in what I hoped would happen. Basically, these resistors triggered the module into sensing adequate resistance and hence supplying the proper 12 VDC to properly light both the map lights and the dome when selected. Everything seems to work properly with perhaps one exception: the dome light glows very lightly when both map lights are on (which probably could be remedied with another similar resistor setup...but I only had two).
Friday, I ordered some similar 150 ohm resistors to further reduce heat buildup in the circuit. Unfortunately I do not know anything about the "module's" specifications regarding min/max resistance to determining when it supplies [full] 12VDC. I did disconnect the quick coupler and measured the two positive power leads and one ground supplying the map lights. Unplugged and with ignition on, one of the positive leads measures 4.55VDC continuously and the other measures 4.9VDC also continuously. In speaking with the mechanic, he seemed to think one of those (perhaps the 4.9VDC) is one that supplies the 12VDC when proper resistance is sensed and lighting is switched on (i.e., door is opened, remote unlocked, map light(s) switched on, or all-interior-lights switch set to on (all three lights are controlled by on/off by a specific position using the same switch as the instrument dimmer switch...many cars do this). Note, however, the dimmer switch does not impact the intensity of the interior map or dome lights whatsoever--again common in most if not all vehicles.
Conclusively, this is all the additional information I have since posting this question and reading your responses--which I'm still digesting. At this point, I'd prefer to generate as little heat as possible with two installed resistors (note: this is why I switched to LEDs to begin with) but also I'm concerned with risk of overheating the resistors if they're left on indefinitely accidentally (although my guess is after so long, the module or computer will attempt to disable the circuit?...guessing here).
Ultimately, I'd like to build a better mousetrap from here. I just wanted to get some response back (and thanks) to those responding so quickly.