I've been experiencing the same problem for several weeks now. I've had my HP Envy and Philips 28E2 monitor for several year with no issues until recently.
I tried turning Bluetooth off and on as you did and got the same response - cycling caused my monitor to cycle between my settings an gamer 1 option. I went to the Bluetooth devices window and noticed that when I turn Bluetooth on I get a notice of "generic monitor (PHL 288E2)" as an available device in the Audio section. Since the monitor does not have speakers I quickly selected "remove device" and the screen cycling did not occur.
Hope this observation helps in some way.