Which Integrated amps should I audition?

It would be wise to determine the new mini-monitors first then only the amp to match, whether you need high-current amps or low-powered amps which in turn depend on the impedance and sensitivity of the new speakers.

I concur with Ryder. Choosing the integrated amp first and speaker second is the wrong order to make this decision.

Jeff Rowland has designed some of the finest integrated amps ever and represent a tremendous value on the used market.

'I" would just wait. If music is at all important, and you do not even know what the new place looks like, wait!
You might find a way to keep your stuff.
I used to live in a tiny efficiency with big speakers and lots of separates. Prioroties.. and, you might find a perfect location for your stuff.
Once you are there, and then you can better decide what is a good option

I have been using integrated amps exclusively for 10 years. I don't think you compromise on quality at all, up to say $20000, even beyond. Instead you get more bang for your bucks. The days of the integrated as an afterthought in a manufacturers range, are long gone.

I agree with Elizabeth, get into your new place and audition in the new room, if possible, with the new speakers and I would say get the speakers first. The critical thing though is sensitivity and impedence of your new speakers. As a rule though, good standmounts tend to be harder to drive, with lower sensitivity. There are'nt that many 96db sensitive standmounts.