This may help:www.seat61.com/Motorail.htm
As Wear The Fox Hat
says above, Dusseldorf is the only place reasonably close to take you to Innsbruck.
I dont know if you have driven to Austria before, but I do it every year at least once, usually twice. It is a fairly straight forward drive. We generalize (bad spelling) that it take more or less 12 hours from Calais to get to Kitzbuhel allowing for a few deisel stops / toilet breaks / coffee etc. That is on the basis of at least 2 drivers; one drives while the other sleeps. With a small child, my brother did it comfortably in 14 hours with feeding / changing stops. Given the location of Schladming, it would be a little bit less but not a lot.
Also, we always drive through the night. This means that the more northern cities are reasonably quiet, i.e. Cologne, Brussells and Frankfurt. Unfortunately it does mean hitting Munich at 12ish which is always rammed and slow. But if you expect that, it is not bad.
If you wanted less traffic, you could head down France and across. The SatNav has highlighted this way. I have never took this route because I am a tight Northerner and do not want to pay for tolls! SO, I can not comment on how easy this route is.
Hope this helps.