Soooooooo......welcome to the land of compromises.
There is no perfect option that will be ideal for everything you want to do. But you can get something that will work in all scenarios.
If you're going to be doing overnights, I really recommend you get some kind of double boot. They're just warmer. The removable liner of course is also a bonus since you can take that into your sleeping bag, but the extra warmth is pretty important too. You do a lot of standing around as the temperature plummets when camping in the winter. That's when everyone gets cold feet.
Also, the routes you're talking about doing are in the grey area between hiking and climbing. I strongly recommend you get something with a very stiff sole. This will give you greater stability in semi-technical terrain, and you can use step-in crampons, which are both faster to put on and more secure. Speed and security are both important when you're in challenging terrain in challenging conditions.
With that, I'd recommend a plastic double boot if you can find them (Koflach Degre or Scarpa Inverno), Scarpa Phantom 6000, or one of the La Sportiva technical marvels (Spantik or Baruntse).
If you opt to go with a single boot, there are many more options, but be aware that none of them are as warm as the double boot options. Also, for overnights you'll need to either take steps to prevent moisture from getting into the liner (i.e. use a vapor barrier liner), or take the boots into your sleeping bag at night, or accept the fact that your feet are going to be truly freezing when you first put your boots on in the morning (which may be either an inconvenience or a serious problem, depending on the circumstances). Scarpa, La Sportiva (Nepal Cube GTX is a perennial favorite), and Lowa (the Mountain Expert GTXs you linked are well-regarded) all make great boots. This style of boot seems to be the go-to for the serious ice climbers I know, though plenty of people climb in double boots too. Rainier in the summer is a little tricky, since you'll have a long, easy approach followed by some steep snow climbing. It's not usually that cold (relative to NH winter conditions), so a single boot works ok. But plenty of people do that climb (and similar climbs like Shasta, Adams, Hood, etc) in double boots too.
Be aware that conditions, rather than the actual climbing, are often the biggest challenge when climbing in the Presidentials in the winter. Depends on the given day and the given route.
Now the real hard part: figuring out how to try a few of these on...