I’ll try and be helpful. My main system is powered by KT150s and my headphone system by 300B. I have two separate systems because they have different purposes and seldom will a tap on a integrated be up to the same standard as the speaker outputs. Also, I wanted a 300B system and there is no way to power the kind of main system I own with 300B… just not enough power.
Having said that… while the tubes used do effect the sound, so does the overall design and objectives… probably more so than the tube choice. The sound is soooo influenced by each and all components that, in general limiting yourself to a specific tube is really making an already difficult choice more difficult. that said KT88’s are good tubes for a tube amplifier. KT120 and KT150s were specifically designed as high end tubes with high output (maybe KT88’s also… I don’t know). Lots of great amps use these tubes.
I am lucky enough to have a great main and headphone system so I was able to optimize both, taking advantage of 300B in a very high current amp design for only headphones. This is ideal… not everyone can do this.
‘’A great sounding main system requires tremendous effort to achieve great sound and stay affordable whatever your budget. I guess I would not compromise your main system to serve the dual purpose of a headphone system. I would get the best integrated amp for your main system. I would go for a Audio Research vsi75. I love it’s warm but detailed musical sound. I would aLso look at VAC, Cary, and Conrad Johnson. If those are out of your price range then Rogue is a good more budget oriented brand. For headphone Woo! I searched many companies and have now owned three Woos… incredible headphone amps! You could use the preamp out from your integrated to go into a 300B headamp and get the best of both worlds.
Good luck.