I have used Gerbing products for many years now and have the set up you describe, plus I have one of their heated liners as well.
My answers are below your questions and a recommendation at the bottom of it all.
1. Will the splitter do what I think it will? (i.e., the lead from the gloves and the lead from the socks get combined into 1 plug, which goes to the bike)
Yes, if you want to. Otherwise you can plug them in separately, although I would not use it this way.
2. What Gerbing product do I need to actually plug the heated clothing items into my bike? From Gerbing's site I see they make an accessory cable, but their language on its intended use is unclear (to me) about its application to BMWs. See "accessory plug" (first item) on this page. Is that what I need?
The plugs you can get from a variety of places, besides Gerbing. Below are other sources you can compare with:
3. I don't have a heat controller yet. How big of a deal is it to install a permanent controller on the LT? I've seen some photos of your results from my search. As simple as this may be to most of you, I'm reluctant to get into it -- just don't have the background, tools, desire to drill a hole in the wrong place, etc. That said, would someone (other than my dealer, and hopefully within a "reasonable drive" -- whatever that is -- to Columbus) like to earn some tax-free $$ by being hired to do this little project?
It is not too difficult to install a permanent Heat-Troller on the LT. However, use a portable Heat-Troller and you don't have any installation issues. I've used the Heat-Trollers for many years without any issues.
4. What controller should I get, assuming my wife/passenger will eventually want her own heated clothing?
Personally, I would not get anything but the Heat-Troller. I think Raffy, who is part of this list, sells it/them.
He also have the new electric liners from the same company. I am using their new liners and the Gerbing liner is now retired and kept as a "back-up".
The new liners are also available in a model especially cut for women and my wife is really pleased with hers.
Best of luck,