Yes You need to experiment to see if its the app (try uninstall & reinstall) or headset thats causing the problem.

Try a dual stream Bluetooth adapter to connect more than one headset or speaker.

You could try a dual stream Bluetooth adapter for iOS products.

Try a Bluetooth adaptor transmitter plugged into or 3.5mm port

You need to use the specific cable for your model iPhone and model TV for this to work. Also it is not a exact mirror of your iPhone it uses only photo apps and video apps. Snip

Apple TV supports 802.11b, g, & n). For music streaming, 802.11b might be OK, but g or n are necessary for video streaming. & naturally, transferring knowledge to Apple TV will depend greatly on your connection speed. Apple TV Features

I got tired of the earbud wires getting caught on everything so I went with wireless earbuds. Sony makes some nice ones and they come with a bluetooth adapter if you don't have one on your phone. It's pretty easy to link up to your phone. here's a tutorial How to Make Wireless Earbuds

Now I know you can only connect to one Bluetooth device at a time, but if you use the audio port of your device (Phone, laptop, iPad) you can use a 3.5mm splitter and put a Bluetooth transmitter on the end of each splitter. Thus, receive sound output to two Bluetooth headsets at once. How To...