we drove down to Lake Garda 4 years ago, admittedly without the van.We travelled via rheims down to L Geneva down to Chamonix, thru the tunnels to N.Italy across via Turin, Milan etc. and arrived and southern tip of L. Garda and then up the eastern side of the lake to Malcesne. Couldn't help noticing the campsites beside the lake of which there were many were very popular with people from Austria and Germany as quite close for W/Es but were densely packed with small pitches on top of each other.
We returned via the Dolomite mountains to Cortina , Brenner pass to Innsbruck,Inn valley to Basle, Mulhouse and back north to the channel.
Wonderful holiday, driving no problems.
We have repeated the trip since with variations, this year are planning something similar with the van on the back.
As someone else said don't travel down the western side of the lake with the caravan, the roads are narrow with many tunnels and no place for a van! you have been warned!!!