Love the discussion here, I think we can probably all relate to one part of it or another.
Your question was about the best expedition rig, but it's a pretty short step from that to "best 4x4", and based on your definition of 2 days, 35" tires, etc, more like "backcountry 4x4 camping rig".
I know that I love bench racing (and building) as much as anyone else, but I have to continually remind myself to just GO. I have been jonesin' for a 4-door wrangler since they came out and I rented one in '07 and spent 3 days on the White Rim Trail and LOVED it (the jeep and the trail). And that was a rental with just 33x10.50 AT's. I vowed to buy one when my eXcursion was paid off.
However, by the time that happened last year, I had a realization that while a JK would be a great all purpose vehicle for me, I already had two really good, capable backcountry vehicles and I should just use them. I also realized that for all the backcountry expeditioning that I thought I'd do in my '99 burb (one of the main reasons I bought it), I never really did, cuz, well, I just didn't. I did other stuff.
We went rock-hounding in the burb a couple times, and to 5 mile pass once, but never multi-day "expeditioning". However, we did go on lots of Boy Scout campouts, and many trips to our mountain lot and lots of tow-the-jeep-somewhere events. My eXcursion took over burb duties in '05, helped build the cabin, and continued to be the vehicle of choice for many Scout and High Adventure outings, family trips, camping trips, rock-hounding trips, jeeping trips, etc. In '08 the rebuilt scrambler not only became better offroad, but onroad too and became a daily driver much of the year.
I don't have a vehicle that is as all-purpose as a JK, "Abner" or many of the other rigs mentioned here and owned by RME members, but I have two vehicles that are more specialized and work great for me. Hmmm, actually, technically, four (five?) vehicles, each with a specialized purpose (more or less). Are any of them expedition rigs (as defined above)? To me they are. And there's not a single RTT. Lots of D-ring shackles though, and 6 pair of tire chains. Here's the list:
F150 Supercrew work truck (not mine, but driven daily, trips to surrounding states, hundreds of dirt miles)
eXcursion: diesel, 8-pass, 35's. It's probably seen more miles of dirt and snow than my jeep has, and way more camping nights.
Scrambler: trail rig, rock crawler and daily driver, when it's not wounded.
Blazer: carries a 7' snowblower in the winter and is a fun rig for a 16yo in the summer.
old F150: RainbowRelicRig for multiday campouts in the 70's.
Is there a single rig that can do all that? Maybe a Raptor...
The ONLY thing I don't do now as much as I did when I drove my sammy and my cherokee is randomly explore roads when I'm on a trip somewhere. I have a few times in the eX, but it's not quite the same. Mainly cuz it's huge.
Last ramble: seems like a third vehicle is an easy solution, is there a reason you haven't done this already? Having a wheeler that you don't have to drive to work on Monday is a HUGE stress reliever. It made me crazy when I was wheelin' my cherokee, which was my only vehicle (besides my wife's car). I remember making the decision to go with a less "all purpose" vehicle and commit to more of a trailer queen (any jeep with 38's isn't gonna spend much time on the freeway). Even if I never trailered it, I'd probably still have a dedicated backcountry rig. It'd just have 35's instead of 39's.
YMMV