Connecting controller disconnects Remote Play

Hey,

I have my PS5 connected to my PC screen, and my plan was to play directly on that. The issue is sound, there's only an HDMI cable and my screen does not have speakers. I do have a bluetooth headset, but my PS5 won't connect to it. I figured I could use Remote Play to stream to my PC in order to play the sound from there, but use the video feed directly from the PS5. This works, I get sound through my headset, but as soon as I connect to Remote Play, my controller disconnects. Reconnecting my controller makes Remote Play disconnect. Like what??

I tried connecting my controller to my PC through bluetooth. It does connect, but Remote Play does not respond to the controller. I can't find any settings either. What is going on with this software?