Bluetooth is extremely hard to get the detailed specs you need sometimes. There are so many revisions.
In the end what you need to get more coverage is stronger transmitters. Bluetooth is very weak compared to wifi. It was always designed to only be a very short distance connection method.
If I remember correctly there are 3 power levels it can run at but it is hard to find that data many times. Things like pc cards tend to have the highest ones BUT that is only 1/2 the connection. Anything battery powered tends to have the lowest power transmitters. This is especially true of earbuds. So the signal might be able to get form computer to your location but the signal going back would not be strong enough.
The problem is the earbuds do not have a good enough radio.
What is somewhat confusing is what you are doing. I could see if you were just listening to audio remotely but you mention a keyboard and mouse. How are these connected to your computer and how do you manage to get a monitor connected in the remote room. Are these running over ethernet or something.
So if you were to have some kind of wire, like ethernet, coax or even powerline networks, you could get a USB extender that runs over ethernet and then plug a bluetooth transmitter into the extender. That way the bluetooth signal would only be in the remote room and not trying to all way back to the pc via wireless
The reason I did this because my room is non air-conditioned so my room would get very hot above around 35c if I'm gaming. That's why I put my PC in the other room.
I'm also doesn't want to use a headphone for the same reason, I don't want my ears getting sweaty.
I connected my monitor using 5 metres HDMI extension. That's included USB extension for my keyboard and mouse but I'm using cheap extension so I think this is the reason why my keyboard and mouse would get freeze if I'm using my earbuds via Bluetooth and gave me a notification saying USB not recognize something.
I did tried using a Bluetooth adapter via my USB extension and it still give a same result, audio signal kept dropping and it would disconnected even I put the adapter in front of me.
I will try to do what you suggest using USB extension with Ethernet and buy a good Bluetooth adapter.
Thanks for your input, really appreciate it.