Hello Peter
I have just noticed your post. I have done the very thing that you are considering but it was over 20 years ago or more.
I spent 2 weeks in Kitzbühel and I took my road bike with me. I stayed in a pension type guest house and the owners were great and told me I could keep the bike in my room provided I did not get any oil on the carpets, so I used to oil the chain on the bed

Just joking obviously!
To be honest I am not sure I could offer you any advice other than to say just go, you will enjoy it. It was so long ago that I can't really remember specific places or routes that I took. I mostly just headed out the door in the mornings and made it up as I went along. When I was there the roads were in good condition and the traffic was never too heavy. I believe that trucks are not allowed on the roads at weekends (or maybe just Sunday, not sure) The main roads took me to most of the places I visited and they mostly followed valley routes. If you want some climbs just turn off the main roads, the smaller roads tend to lead to smaller villages etc and some lovely views can be had. Some advice I can give is just the obvious sort, if you do venture off the main routes do have a good map with you. I remember one ride I did by just trying to follow road signs. I followed a sign that took me away from the main roads but still pointed towards Kitbühel, it was a lovely single track road that climbed and climbed, it was a beautiful hot sunny day and the views were very rewarding. However, at the top the road became a track, which became a footpath through some trees, which then became a dense forest without any signs. By now I was having to push my way through thick branches and then drag the bike behind me step by step down a very steep and rough descent. I was soaked in sweat, had my own private swarm of insects buzzing around my head and was teetering around in my cleated cycling shoes. It was at this point a man appeared from nowhere, dressed for a days hiking. He was Austrian or German I'm not sure, but he stopped, looked at me and simply said "English?" I nodded to confirm his suspicions and he just walked on by shaking his head. I eventually "fell"out of the forest onto the road somewhat dishevelled, scratched and bitten. A great day!
It was a great holiday and I fell in love with Austria and have been back many times, although I never took my bike again. I may have been lucky but on all of my visits the weather has always been sunny and warm to hot. I have never been disappointed with any of my holidays in Austria, in my experience it is spotlessly clean and just a lovely place to spend some time, whether riding the bike or walking, and taking in the spectacular surroundings.
A couple of places have come to mind that I remember passing by or through. There was the Krimmel Waterfall or something like that, although I did not stop. One day I seem to remember coming back through Zell am See, I have since been there on holiday and it is a gorgeous place. If I remember anything else I'll let you know, I will dig the maps out and see if anything jogs my memory.
Please let me know if you decide to go, I would be interested to hear how you get on. It goes without saying that Kitzbühel is a lovely place to stay too.
Good luck
Dave