I did a similar route in reverse last September, travelling from Hungary, where we now live, to the UK. This was our first big trip in our newly acquired Eriba, although we had done the journey many times in a motorhome it was a steep learning curve for adapting to towing and living in a "caravan".
We used motorways for speed and convenience, tolls through Austria which can be purchased at a service station when filling up. No tolls through Germany of course but we did find an awful lot of roadworks on the Autobahn system. We chose campsites for overnighting as close to the motorway system as possible and did the journey to the ferry port with two stopovers. No issues at all with towing, the Eriba really is a dream to tow and even more so with your car I imagine. To cut down on the hassle we pitched without unhitching if space permitted, on our overnight stops, just lowering the rear stays. I noticed a lot of the German "tuggers" were doing this. Make sure you have plenty of cable for EHU as we found some pitches were a long way from an outlet.
If you're coming to Hungary let me know if you would like more country specific info regarding campsites, what to see etc.