The quickest option is through France, past Geneva and through the Mont Blanc tunnel. Still v scenic either side of the tunnel. Mont Cenis pass is worth considering as you save the tunnel toll and the views are stunning.
We were towing (and with 2 young children) and comfortably did Devon to Reims, Reims to Geneva and Geneva to Garda.
The year before we went through Germany (Koblenz, Munich) and over the pass into Italy on the way out and via the Frejus tunnel and through France on the way back. German route was cheap, and was a no brainer as we were visiting friends near Munich but took much longer in terms of driving hours (3 long days from Calais).
We always go through Belgium, Germany to Innsbruck and down the Brenner motorway easy route and scenic once you get out of Belgium.
Very little tolls paid.
The best Campsite is Nattersee situated very close to the Brenner Motorway entrance, It is listed on here in Austran Campsites it is also an ACSI Camping Card Site so listed on their website.
Hope this helps.
DiPhil