Integrated Headphone amp with moving coil phono stage

Look into Linear Tube Audio Z40 and Ultra linear integrated amps. They have a headphone out and the option for MM/MC phono as an add-on.

Unfortunately there’s almost zero market in hifi for the intersection of these things, so anything you come across that gets close is almost certain to use low cost off-the-shelf chip based solutions for most (if not all) of the functions you require:

• Headphone amplifier with low impedance driving capability (32 - 300 ohms) compared to a traditional preamps. The throw-in headphone jacks you see in DACs, receivers, preamps, integrateds (and other such products) are sometimes serviceable, sometimes AWFUL, and almost never near the best of what you can get from headphones.
• Volume with balance control
• RIAA phono stage and MC headamp

If you care about quality (which is presumably why you’re here), you’d be far better off with a dedicated phono stage plus a dedicated headohone amp, each of good quality with discrete circuitry. Schiit Audio makes this kind of stuff with a good bang-for-buck. I would also recommend Hagerman Labs - more for the phono side there, but he does have a headphone amp too. I’m an old head-fi guy and if I was starting from scratch today I’m getting an old dynalo (Kevin Gilmore Dynamic), dynamid, dynahi, or GS-X build, whatever I can find, and calling it a day. Maybe an EARMAX or similar OTL if I want to go the other way and wallow in glorious double-digit distortion. Maybe both! But getting a balance control too is going to be tough, without adding a 3rd box and degrading quality with a cheap potentiometer. 

a full featured integrated amp with headphone jack and mc phono:  this one, top right is balance and volume, not all have balance

https://www.finn.no/recommerce/forsale/item/407968337

Hi OP

This may be $$$ but the Mark and Levinson No 523 has the phono pre-amp. headphone driver and quite possibly the balance control you all needed

If this fits what you need, please check this out.

Mark Levinson #523: 6-input dual mono Pre-amplifier - Stereocheck

you need to clarify certain details with seller, many functions are via remote/menu, not sure if balance is available via the menu

"BUILT-IN ANALOG PREAMP: Line level input or Phono M/M, M/C input, relay controlled."

https://www.canuckaudiomart.com/details/650234493-brooklyn-dac-dac-preamp-headphone-amp-and-phono-stage/

asking $850 CAd = $615. USD

different listing, here, may learn from it, BBII, a lot more money

https://www.audiogon.com/listings/lisbej5f-mytek-brooklyn-bridge-ii-all-in-one-roon-core-streamer-dac-headphone-phono-preamp-e-da-converters?refsource=hifishark