I've done it unsupported/camping on a mountain bike (slick tyres) and supported on a carbon race bike. Ironically the unsupported version was the shorter tour.
It used to be that if you were carrying luggage then you generally needed a frame with lugs for a rack/guards so you were restricted to touring style/audax frames, however the market for frame mounted luggage is now huge and you'll generally find something that will fit on whatever you have. Lots of the new adventure/gravel style frames have every luggage mount available on them and may be up your street - but you may want to swap out the tyres for something that leans more to on-road than off-road (presuming that's where you'll spend most of your time).
However, you don't particularly want to buy a bike for the purpose of a two week tour. Buy a bike that will suit your cycling requirements for the next few years - as long as it can be adapted for your touring requirements. Otherwise it won't be much use once you've finished.
Ultimately you need something that you'll be comfortable on for 70 miles a day for two weeks, with wheels strong enough to hold the weight of you, plus the bike, plus the luggage you carry.