The simplest solution for you: Watch whatever you want to watch and listen that way on the Roku you already have.
The simplest workaround for when you want to use the AppleTV instead: Presumably this is not something you want ALL of the time- just some of the time. So in that "some," airplay whatever you want to watch from your phone to AppleTV. Since your phone will be the source of what you are watching, the buds you have will work and take phone calls too. Bonus with this is that if you are out of sight of your TV as you are roaming around and you hear something that makes you want to see the video too, you can watch it on your phone vs. going back to the TV room.
If this is a frequent want, post #10 is probably next best option for when you want to use AppleTV instead of Roku.
If your AppleTV is pushing audio through an AV Receiver, you can probably connect buds to the receiver to facilitate this want. If no receiver, there are bluetooth transmitter devices for relatively cheap: AppleTV to Transmitter, synch buds to transmitter. However, both of these 2 options face the same issue of having to choose to which device you want buds linked (phone vs. receiver/transmitter) unless you get some dedicated buds/phones for this option (#10 again).