Can you rule the Austrian Empire from a non-Austrian city? Is there an Austrian city left that would be able to serve as a replacement? I think the point that people are making who are unhappy is that the answer to those two questions is no and no.
I found this link that contains the population of Austrian Empire cities in 1850:
https://www.reddit.com/r/MapPorn/comments/gtfw1j
Assuming non-Austrian cities are out, then the new capital would be Graz, right? With a population of 55,000. Could Graz serve as the capital to the Austrian Empire?
That said, I think it is important to recognize that Austria is a bit unusual. Its multicultural nature makes it the GP/major power that would be the most devastated by losing its capital. Most GP/major powers have large homelands with cities that were important and could serve as an alternative capital. So, Austria is an outlier.