So have just done this - in fact, do it regularly, but this was my first trip in the van.
I chose to go via Fuessen this time -some hints on the trip down if you require.
1. Get across Belgium & Holland to Germany, cross near Cologne. Be aware in most German cities now you need a green TuV ULEZ sticker - best to pick it up over there. You can search Google for TuV garages, you just present then wth your V5 and 6 euros later you have a green stickr that will never come off.. If you want to break the journey before the Rhine, Leuven near Brussels is a good alternative to the usual Bruges or Ghent. Or Valkenburg near Maastricht.
2. I stopped at Lorelei (great campsite with views), dropped down the Rhein and into Heidelberg but then went on to Rothenburg (recommend Camping Tauber-Idyll). Heidelberg camping is not brilliant tbh and parking around the city is a nightmare. But if you have time, you can stay near Weinheim (more North) and just get a tram in, lot less hassle.
3. Then on to Fuessen for an overnight or two by the lake. Also stopped in Kempten on the way back, which is OK but not as pretty.
4. Driving over then to Innsbrueck and then over again the Brenner to North Italy, toured around Lake Garda. Note for Austria you now need a Vignette for motorway travel (book it online the night before, you can get a 1-day or 10-day pass) as well as the tolls for the Brenner.
This route will take you 3 overnights if you are hell bent on getting down there. German roads are fast, kee pan eye out for campers with Dutch (yellow) or Belgian (Red) numberplates, they're always a nightmare on the autobahns.
Overall, it's a lot longer and a bit more hassle than my usual route - straight down to Luxembourg avoiding the French tolls (1 night stopover round here somewhere), then (after filling the tank) down through Strasbourg/Colmar to the Black Forest/Freiburg and then onto Bern/Interlaken (2nd night) and over(actually under) into TIcino via Gotthard (Ancona is quite nice on the lake there). Alternatively, you could head over to Chur and take the Julianpass over to St Moritz, and thats also a really nice drive. Switzerland is obviously more expensive but the journey ends up being a bit cheaper on diesel & tolls even though you need a vignette for Switzerland. Plus camping on the lake at Interlaken/Thun is great.