You are very lucky to have the extra week! I am very lucky to have an understanding wife (and sympathetic kids) who doesn't mind when I take a week for these kinds of trips on my own.
While mountain bikes are pretty popular for bikepacking in more remote areas, you'll see very few of them used for road touring in North America. As you mentioned, the geometry is less comfortable for that style of riding, but the thing that will get you frustrated faster is the speed (or lack thereof). Yes, you could swap out your knobbies for slicks, but you are still riding something geared for torque. So... you probably want a different bike.
Before you take that as license go all fancy, recognize that (1) you may find you don't like bike touring, and (2) you don't actually need a fancy bike to tour. The former may not be a real thing, as I have yet to encounter anyone who doesn't love touring if they are already keen on cycling. The latter, though, is a real consideration. The truth is that pretty much any mechanically sound bike can be used for touring, though a mechanically sound road bike will be more pleasant. I did multiple tours on my 2004 road bike (not a tourer) before I invested in a true touring rig.
My advice would be to look for used bikes that you could tune, if you are decently skilled, or have tuned for you. If your city has a bike co-op that sells refurbished bikes, that is likely your best option (cheap and pre-tuned; mine sells great bikes for <$300). Then you will need to add some $$ for bags and such, but I would stay minimal as possible and not end end carting around the kitchen sink.
With all that said... an expensive bike you never ride is a waste. You could sell your MTN and buy a decent tourer (Surly LHT is one of the best budget-conscious rigs available in NA) that could double as your commuting bike. I use my tourer for commuting, and the ability to amortize the cost of the bike and my bags/equipment over both touring and commuting made it nearly a no-brainer for me.
