Innsbruck to eventually cavtat

I do not understand the first part of the question, about what is logical after Innsbruck.
Three days is not a lot of time to traverse Croatia. You can take a train to Zagreb, and after that--what are your interests? Are you open to renting a car, or would you like public transport suggestions? One option would be to stop at Plitvice and then at Split before continuing on to Dubrovnik/Cavtat.

Berlin to Munich to Innsbruck to Ljubljana.
What method of transport were you thinking of? "We would like to avoid long bus journeys if possible" - buses are generally slower, less frequent and a lot less comfortable than trains. All of the above legs are reasonable by train, the longest is Innsbruck to Ljubljana which is 6-7 hours.
Cavtat (not Cavat) is more difficult, I had to look it up! It is on the Croatian coast near Dubrovnik. There are no trains in that area. https://www.rome2rio.com/ suggest flying to Dubrovnik then a taxi.

I had no problem with the buses in Croatia. I think it depends upon whether you have an interest in seeing some places other than Cavtat/Dubrovnik, because that's a long slog of bus rides if Cavtat and Dubrovnik are your only goals. Zagreb, Plitvice Lakes National Park, Zadar and Split are among the worthwhile stops.

Another possibility would be to take the train or bus to Zagreb (a city I really like) and pick up a rental car there, dropping it in Dubrovnik.

Valadelphia, thank you for your reply. The next logical place, meant from looking at the map, it appeared to be a lovely destination and on route to Croatia. We are open to hiring a car, but thought it may be very expensive to rent in one country and drop off in another. We are just very unfamiliar with this part of Europe.

Ljubljana is a very attractive city with a very nice vibe. Much more intimate than Vienna, for one example. Built along a narrow river so you can zip back and forth easily, which I really like. Lots of cafes and restaurants, good food, a couple of art museums, a castle (not a mandatory stop in my view; it takes a good bit of time), and not expensive. Plus there's interesting architecture running across several centuries. You'll enjoy it.

It's highly likely that renting a car early in your trip and dropping it in Dubrovnik would be cost-prohibitive, but picking up in Zagreb might not be. Be sure the plan to cross a bit of Bosnia-Hercegovina is noted, however.

Not too many Americans get to Zagreb. Too far from the coast, I guess. But it, too, has a very nice historic district, and quite a lot of interesting museums. Active street life, albeit without the river. I should perhaps warn you that Zagreb is fairly large and sort of sprawls. I didn't observe anything that made driving there look especially difficult, but it's not a tiny town you can just drive through in 5 or 10 minutes.

Both cities are very easy to visit since English is so widely spoken by the young people.