List of tube integrateds with headphone out

All of the "golden age" tubed integrateds circa 1955-1965. When refurbished these can perform incredibly well. The issue for you may be the output power. You might want to specify which speakers and headphones you want to drive.

The Fisher Receiver I am using sounds good to me. I believe the preamp circuit is used to amplify the headphone jack. The tubes are usually NOS German.
However if you want a super headphone amp I will admit that my modern but now disgraced SinglePower is much much better. If you audition a quality dedicated headphone amp I bet you will be bowled over if it sounds anywhere as good as the one I have. (Despite the bad behavior of it's maker.)
I must apologize I do not know a current tubed integrated as specified.
I wonder given the nature of this industry if you can ask for a custom piece can be made for you. Mapletree Audio in Canada makes preamps, amps, integrateds and Headphone amps usually made to order. He seems to be flexible but I haven't any idea how much it would cost.