I haven't tried a tube amp before so this is new territory for me.
Consider an amp that can use more than just EL34's (good start IMO).
Also consider that with an auto-biasing amp if there are more than 2 output tubes you should really stick with matched quads. Meaning if you lose a tube a replacement may be too different from the other 3 and the auto-biasing may not be able to handle it. With some tubes pairs are a lot easier to find especially with vintage tubes.
Last consider an amp that has great protection circuit for the output tubes. Without it depending how a tube decides it's time for tube heaven it could take out some resistors. Also a soft start circuit helps extend the tube life.
That being said I haven't seen anything that is all of the above anywhere near your budget. I have an Octave V70SE integrated that does all of that but it is 7k (have only seen 1 V70 not SE used for 5k). So consider these when looking.