Did you say what month? It makes a difference.
My view is what can be done and what is enjoyable are two different things.
Are you purely going for the riding or for a nosey round also?
I would pick one or the other, both would be an endurance test.
Austria will feel further in terms of getting their, it always does to me.
Innsbruck is a lovely city though. Many passes charge to cross if that bothers you at all.
If going to Switzerland you have to visit Lauterbrunnen valley, you need days there to appreciate it. The main passes are a day ride from there so it makes a perfect base.
I am biased as got married in Interlaken and we booked out the Piz Gloria revolving mountain restaurant for the reception. There are many great places to visit.
Time of year matters though as in high summer it has become very busy.
June is better, passes tend to open around the beginning of June but can be as early as late May or late as second week of June.
Switzerland can be done without a vignette for the motorway but just buy one as even the motorway is interesting (and I hate motorways)
