Opening an LLC is perfectly legal even though you may not live in that particular state, Trucking companies from all over the country do it all the time, look at the tags on many tractor trailers on the highway, a very large percentage of them have Maine tags, even though the company may actually be from VA.,NY, or anywhere else, WHY, because taxes & tags are cheaper, but in order to register there as a non resident, you must open an LLC, its not just the OP looking to get out of paying taxes, he's going to pay taxes where ever he lives.
Now, after that, I'd say that it would be easier to get a 30 or 90 day temp. registration, (IF CT has such an animal)then when you get to MT you can rent temporarily, or get a seasonal site, which I would think would still be renting & therefore wouldn't need to buy a house right away, but it would make you a resident, in which case you don't need to open an LLC, just use your renting address to register your vehicles.