You might try booking.com. Use all the filters you want. Then you will have to scroll down the page to find the pet policy.
I recommend you ignore the stars and use the ratings instead. Personally, I only consider 8+ properties. In either case, rooms in the center won't be cheap.
Some places have hypoallergenic rooms. I saw a few others that didn't allow pets at all. Just to be sure, contact the hotels that interest you directly to verify how they handle pets. Sometimes there are only certain rooms where pets are allowed.
By limiting to the city center, you severely cut down your options, so you might want to broaden your search.
Finally, if you haven't been to Austria, it is a very dog friendly country and it's quite common for dogs to be allowed on public transportation, in shops, and in bars and restaurants, sometimes inside and definitely outside. You will probably see lots of them in June. If you are highly allergic to dogs, you might want to consult your doctor about meds for that.