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I used this blueprint, it allows you to use the buttons for turning on and off, hold to increase/decrease brightness, double tap for full or minimum brightness.
Getting the brightness to smoothly change upon holding the button was impossible for me to figure out on my own, this blueprint really saved me in that regard.

The double tap changes colour temperature when you use the button connected directly to the lamp, which this blueprint unfortunately does not replicate for HA. I have yet to summon the courage to really go through the yaml and see if I can figure as much out for myself.

You can go to this website to see if it’s supported, though it seems you’ve already figured out it isn’t.

https://test-ipv6.com/

Fun, I just went through this with a Bilresa remote.

IPv6 needs to be enabled on your HA server, I ran this command though I’m not entirely sure if that was required: ha docker options --enable-ipv6=true

But what really seemed to do the trick for me was, on my phone, going to Settings > Companion App > Troubleshooting (almost near the bottom) > Sync Thread Credentials.

Why there? Good question!
Why not the “Send thread credentials to Home Assistant” button at the actual Thread configuration? Equally good question!

This was with the Home Assistant Green and ZBT-2, mind you. I would’ve expected there to be some level of “out of the box” integration. Oh well.