Suggestions for a tube integrated under $3000

Some good recommendations already. Have you thought about something like the MastersounD KT88 or EL34 amps?

A real bargain would be the Aronov LS-960I if well maintained, and about $1200 pre-owned.
It has a most natural, musical character, together with the best ability to optimize bass accuracy and extension anywhere near this price range.

buying a Cayin or VAS is a must, great sound, great value. I own four of them, never a problem. Consider buying used if you can do a pick up, for cash.

I've heard the SLi-80 and the Prologue II on those speakers . I much prefered the Primaluna over the Cary due to its better extension on the top .

Good luck

There is only one I would buy new at that price - decware flagship - zen Torii mkiii. Amazing bang for the buck. Buying used I would look at CJ ca200 or Pass labs class A 30 watt integrated. I have a Simaudio i5.3SE I would sell for $2500