First, I feel for you. This isn't an easy situation. I get exactly what you are after.

Audio is definitely stripped off the HDMI connection at the receiver. You may have some option in the receiver to pass audio onto the display 'all the time', but then the audio will be converted to stereo, I believe before the receiver. Not sure about that. That may be a good question for the manufacturer. But, check the settings. It also may be worth asking at a place like Audioholics or AVS Forum where people who own the receiver may be able to give you more specific insight and maybe an option I haven't thought of.

One of those options...

You might want to consider a different options for your whole house audio streaming solution as an idea. If you get a stand along streaming audio player, like a Sonos, you can use that to feed your whole-house audio system. It has a top-shelf phone-controlled playback mechanism and may be ideal to free up zone 2 of your receiver to allow for you to connect headphones to.

I am not sure the latency of modern Bluetooth. Many people do game using it, but I can't talk to much one way or the other. I just know that I've picked up the Sennheiser models and probably have half a dozen customers that really do like them a lot for comfort and usability. They have a local on/off switch as well as local volume control which is nice. They are analog instead of digital, so they sometimes get a bit of interference, but for hearing what's playing on the TV at a volume you like, they are spectacular.

I use a more expensive set of digital headphones from Sennheiser, they do sound better, but aren't perfect.

I may pick up a Bluetooth Tx unit like you listed just to give it a try. I'm sure there will be some headache about it which isn't obvious.

A big plus to the dedicated headphones is that they tend to 'just work'. That is, you pick them up off their included charging base, put them on, hit the on switch, and you have audio. Bluetooth sometimes requires manual pairing.

If you do post elsewhere, let them know that you are using zone-2 for whole house distribution so they don't keep wondering like I was.

Good luck. If nobody is responding on Audioholics, then that sucks. Often I get tagged in those posts and I end up involved.

A three zone receiver would do the same thing you are looking for, but that would suck to have to buy.

Sonos is an app-driven music player like a Roku or Firestick is. You load music apps onto it and it just works. That's something you can ask about as well. I have a Sonos, but there are other things that act similarly. If you don't need cable TV or other sources from the receiver then a stand-alone dedicated music player may work really well.

I use my Sonos on a hard wired network, it has an integrated network connection.

I'm not sure what you mean when you say you just put in a URL on the Denon. I would be willing to try something direct using Sonos to see if that works. Sonos is pretty 'premiere' as a product, so they have extensive support and their own forums groups I believe.

Just Googling it...

Sonos does support streaming radio stations. So, if your sole purpose for zone 2 is getting streaming radio stations to speakers, then a Sonos can accomplish the same thing. I would look on eBay for a used model. Either a Sonos amp (if you need amplification for the speakers) or a Sonos Zone Player if you do not need amplification.

Then just use zone 2 for your headphones.