That's complicated things straight away!
The airports in Bavaria are Munich, Nuremberg, Memmingen and Augsburg.
Augsburg no longer handles scheduled flights. Nuremberg is only served by direct flights from the UK by Ryanair, out of Stansted. The same applies to Memmingen. Munich is served by BA from Heathrow, easyJet from Edinburgh, Gatwick, Luton and Stansted, FlyBe from Cardiff, Lufthansa from Heathrow, Gatwick and Manchester, plus Singapore Airlines from Manchester.
Although it's not in Bavaria, Frankfurt is close enough to be considered as well. It's served by BA from Heathrow and London City, and by Lufthansa from Aberdeen, Edinburgh, Heathrow, London City and Manchester. (Frankfurt's 'other' airport, Hahn, is probably less well-placed for getting to Bavaria, and it's only served by Ryanair from Stansted anyway).
Salzburg Airport is another possibility, as it's only just over the border from Bavaria. It's served by BA from Gatwick, Heathrow and Glasgow, Jet2 from Edinburgh, Leeds/Bradford and Manchester, Monarch from Birmingham and Gatwick (but both are 'seasonal only', so there might not be flights in December), Ryanair from Stansted, plus Thomas Cook from Gatwick and Manchester (again, 'seasonal only').
Looking at it the other way round, there are no flights to Germany (or Austria) from Bournemouth. The only flights to Germany from Southampton are to Hamburg and Dusseldorf, which are both on the opposite side of the country to Bavaria. There are no flights to Salzburg and (despite the destination still being shown on the Southampton Airport website) it seems that Austrian Airlines have discontinued their flights to Innsbruck (which might also have been a possibility).
So it seems that, unless you're prepared to take connecting flights, you'll need to travel to a London airport as your departure point. If you'll be travelling by road, Gatwick might be best (as it's reasonably accessible and has budget airlines operating to Munich and possibly Salzburg). If you'll be using public transport then Heathrow might be better (as you can take a train to Woking and then the Rail-Air coach to the airport) but that then means that you'll be stuck with using 'full fare' airlines.
Any preferences?