Is it mandatory to only use alkaline batteries (“LR6” designation) with the Xbox One controller?

From my experience, I use Energizer NiMH rechargeable batteries in my Xbox One Elite controller (and a regular Xbox One Controller as well), and have experienced no issues what so ever. I don't think the material of the battery has any significant impact, if any at all on the controller. Mine still work, and I've had them for years now.

I can't say anything about that particular controller (I don't own one), but I can shed some light on why they might be recommending against rechargeable batteries.

While the controller doesn't really care about the particular chemistry of the battery you put in it, it might care about the battery voltage.

Regular alkaline non-rechargeable batteries come in at ~1.6V fresh from the box and have a fairly linear discharge curve. NiMH rechargeable cells have a voltage of ~1.4V when fully charged and sit between 1.1V and 1.2V when discharging most of the time. Some appliances work great with both types while others refuse to work at the 1.2V level, which means that NiMH batteries don't really suit them - you can't access most of the energy stored. That might be the case here.

There is a third option though - you can look for lithium based rechargeable AA batteries. They have a built-in buck converter and USB charging circuitry. You can get them from e-bay or amazon, though they are more expensive than NiMH batteries, so stick to those if they work for you.

It's not mandatory. That's only a power source. Using alkaline batteries prevents eventual risk of leak if controller lay unused decent amount of time. If you won't let your NiMH's lay discharged a long time that shouldn't be a problem. But even when leak occurs nothing happens... except risk of environment contamination ;)

Edit: NiMH's doesn't like slow discharging, it leads to raise of its internal resistance and consequently faster use (becoming less efficent faster). But if you play intensively from time to time, that's not a problem.