They're on separate ends of the analytical/musical spectrum, and should complement each other nicely. They're also the two most comfortable "summit" headphones available today, imho.
Getting a single amp to synergize with both might prove difficult though, but (like others have mentioned), some of the Japanese Bakoons might be able to do the job. I heard the HDA 5210 sounds great with the TH900.
Can't comment on the LCD-3. I've only tried it for a short time (maybe 3 hours max) and it sounded good with an Alo Studio 6 amp. Problem for me was the comfort. Slightly more comfortable than the LCD-2 (which I used to own) thanks to better earpads, but the clamping was still too strong for my skull, and the earpads were still far from perfect for my ears. Sound-wise, I'd say the LCD-3 is in between the HD800 and TH900. Doesn't quite have the imaging and wide soundstage as the HD800, but it's nowhere near as sterile. Not as bassy or fun-loving as the TH-900, but compensates with some of the best midrange available. Maybe a good compromise, depending on your presentation, as long as your head can stand the force.