To answer your initial question, the mazda 6 would be a far better road trip vehicle than a suzuki swift.
The mazda6 is mazda's largest sedan (the comparable mazda to the suzuki swift is the mazda 2).
I have a corolla hatchback (the size up from a swift / mazda 2), and went to the snow for the weekend with 4 people (total). Boot was full. (we were renting gear too, so none of that)
The corolla boot is much bigger than that in the swift, also not much space at all in the back of a swift.
Don't worry about engine size. People / cargo space will be a much bigger concern for the trip.
Take a serious look at renting a car for the trip if you were buying for this purpose. Using the next two weeks dates (if you are traveling over the normal New Years busy period, prices will be higher), Apex will give you a Wingroad station wagon for $650, or a Toyota camry for $1021, or a Subaru Legacy wagon for $1111. (If you are the only driver buy domestic travel insurance as a cheaper option than rental car company excess reduction insurance.
If you do decide to go for the trip in a small car, consider getting a roof box to handle the cargo, sometimes they can be picked up cheaply off trade me (new ones are around $1000). If you have somewhere to store it, you can take it off the car for the rest of the year to save windage.