We rode from Lyon* all the way to the Med in 2008, using the fantastic Rhone Cycleway for most of it - a tarmac cycle path and quiet roads.
That was in September we had the north wind with us (can't remember what it's called ... "Tram ... " something?) for a couple of days. We were averaging almost 19mph wind-assisted on steel tourers with panniers ... although it was pan-flat other than the odd bridge over a lock!
Having been in the S of France many times in June there's no real pattern to the Mistral ... later in the month seems less likely. IIRC it usually lasts no more than 2 days.
As for direction, I would suggest north-south as being better . . with the finish at the glittering Mediterranean Sea.
Rob
* Having started at Calais - there's a Journal about it at
www.cycle-endtoend.org.uk [and, shameless plug, a book that we wrote is on sale :
Wine-ding Down Through France The ride was inspired by Edward Enfield's book, Downhill All The Way]