I love Paris, but for December it wouldn’t be my first choice. If you have any wiggle room on that decision …….. Still, its Paris!! So I am cutting hairs here.
I would fly next to Prague. Direct flights start at about USD60 so it’s pretty economical and a short flight at that. Then Vienna and then Budapest; but with some stops in-between. That particular direction because I have a bias for Budapest and think its crowning touch to what is a pretty standard tour route. Here is my 3 week suggestion
1 Saturday, depart Australia
2 Arrive Paris
3 Paris
4 Paris
5 Paris
6 Flight, Paris to Prague
7 Prague
8 Prague
9 Prague
10 Shuttle Service, Prague to Cesky Krumlov
11 Shuttle Service, Cesky Krumlov to Vienna
12 Vienna
13 Day trip to Melk
14 Vienna
15 Vienna
16 Saturday, Train Vienna to Gyor
17 Trip to Archabbey at Pannonhalma; then train to Budapest
18 Budapest
19 Budapest
20 Budapest
21 Budapest
22 Saturday, Return home
You didn’t mention being there for Christmas or New Year so the only 3 week period that excludes that begins on Saturday the 3rd of December. If you can be there for Christmas or New year, more the better. I love Budapest at Christmas and the whole region is beautiful and entertaining during the holidays.
None of this requires a car and in fact a car would be more of a burden than an asset. Also keep in mind the roads in December can be less than great.
BUT, if you want to brave it, then how about you move on from Paris to Prague, see Prague for a day then do a driving tour that ends back in Prague. There are a number of 19th and 20th century resort towns and dozens of beautiful castles in the Czech Republic. Maybe 6 days all together. After another day or two in Prague fly to Budapest. Maybe 6 days all together.
In Budapest do something similar; except I would spend 2 days in Budapest then head off through Slovakia to Krakow. Slovakia has castles that rival those in most of Europe and the mountains are beautiful as well. I would spend a night at one of the Banska towns and another further up in the Tatra mountains then on to Krakow. After a couple of days in Krakow head back to Budapest by way of Bratislava and the archabbey at Gyor. This is more of an Eight day trip by the time you add a few more days at the tail end in Budapest before flying home.