The headphone amplifier must has this feature.
There are several analog-only headphone amplifiers that accept XLR inputs and have 4-pin balanced XLR headphone outputs, such as those you linked to. I would recommend sticking with that approach as you've stated you want to keep the analog output of the X1.
That being said, the additional feature you probably want is some way to pass through from the headphone amp to your speaker system. As the X1 only has one pair of XLR analog outputs (unless you use RCA for a second set of outputs and switch between the two when you are switching between headphones and speakers—but I think you'd ideally want to avoid that).
Two units I'm aware of supporting what you're looking for, that also provide an exceptionally clean signal path without coloration, are the OPPO HA-1 (discontinued but can still be found) and the Bryston BHA-1. Some people prefer to get the BHA-1 in a "low output" mode for the rear outputs, as the normal output level is too high for their speaker setup.
If you read some of the posts in the AudioCircle Bryston BHA-1 thread (linking directly to a relevant post) you'll see that the some people order their BHA-1 in a "low gain" configuration, because otherwise the XLR output on the back is too high/loud for their speakers. In the default configuration, they had to turn the BHA-1 volume knob very low otherwise their speakers were too loud.
Thanks for the information about Audioshark site.
I have registered and I am going to try to reach Wklie.
I am wondering if I would have been better of , buying the Matrix Element X instead of the LuminX1. I have read that Tidal , Qobuz ,
etc , will be implemented with a futur firmware updates.
The headphones issue would be solved.
Audiophile regrets ...as usual.