Dolomite Transportation

As a first impression, Dobbiaco looks the best base; I can't give exact details as I was not there for too long a time, but there should be buses both to tre cime and to braies. As for Val Gardena, I am afraid you will need a couple of trains to Bressanone or Chiusa, then a bus; I am afraid you are underestimating this trip.

I think there might be a bus connection (perhaps requiring multiple buses) to the Val Gardena as well as a bus-train-bus option; something I read a few years ago left me with that impression. Rome2Rio suggests I am right. Its suggestions all involve traveling north.

It's important not to trust the details posted on Rome2Rio, because it is often wildly off-base about travel times, frequencies and fares. If you keep drilling down on that website, you can probably find the name of the bus company(ies) involve. Go to their websites for accurate schedule info. Be aware of seasonality (perhaps more service in the winter--for skiers--than in the summer?) and day-of-week variation.

Rome2Rio routing suggestions

At this time it looks that most public transportation in Alto Adige area is administered directly by the province through its mobility agencies that manages both regional trains and buses. If you go to their search engine, https://www.suedtirolmobil.info, you can have an idea of travel options at the present moment (not in next summer). I see that Dobbiaco to Ortisei can be managed in about 2 hours 30 minutes by a combination of train (sometimes direct, sometimes with a change in Fortezza) to Ponte Gardena, and bus from Ponte Gardena to Ortisei.