apart from the added fees, a Czech car may have other problems. Winter tyres are mandatory on all cars in Austria from November until mid April. Czech law isn't that strict so the car may not have them. This needs to be guaranteed in writing before booking. It also won't have an austrian autobahn toll sticker (google Vignette).
Dachau is more than just outside Salzburg, it's a suburb of Munich and in winter conditions would be about 3 hours each way. Berchtesgaden is a local bus or train ride.
Excluding Budapest and Dachau this is fairly easy. And without a car.
Fly to Prague. Be careful with London airport, does the Virgin flight arrive Heathrow or Gatwick? Beware that Stansted is a long way the other side of London, Luton is a drag from Heathrow, fairly easy from Gatwick. And Heathrow to Gatwick isn't as easy as it should be.
After Prague get the train. You can either go and stay in Český Krumlov or České Budějovice (and visit Český Krumlov from there, about 45 mins). Obvious mix is get up fairly early, go to Český Krumlov, having seen it go to České Budějovice for the overnight. Budějovice in pre 1918 German is Budweis, and it's where the real Budweiser beer comes from!
czech trains are cheap (if slightly worn) https://old.cd.cz/en/default.htm czech spelling of Prague is Praha
Next day to Salzburg. From České Budějovice this is easy, you just get the train at 8AM or 10AM to Linz and a fast train from there to Salzburg. From Český Krumlov it's easier to pay for a car and driver, these people get mentioned a lot in this forum https://www.ckshuttle.cz/transports/transfer-from-cesky-krumlov-to-salzburg/ 1000 Czech Koruna is about $45.
Berchtesgaden. local bus line 840 from outside Salzburg main railway station, takes 45 minutes, once or twice per hour.
On to Vienna. Railjet train, book early at http://www.oebb.at/en/
Fly back to London. either from the real Vienna airport or Bratislava, don't laugh - it may be capital of Slovakia but the airport is reasonably handy for Vienna and this is where some budget airlines go to. Details of airport trasnfer from Vienna will be on the site of any airline that uses Bratislava
"normal" airlines, may be dearer but more convenient
czech airlines http://www.csa.cz
austrian airlines https://www.austrian.com/
the budget airlines can be really cheap but checked bags are paid for as extras when booking, and you usually need to check in online and download your own boarding card (print it or carry it on smartphone). And make sure you understand which airport they are using!
https://www.easyjet.com/en
https://www.ryanair.com/gb/en/
https://wizzair.com