I had a similar problem with my Pi4 and a BenQ projector. The projector would only recognize the Pi when it was already running before the Pi booted.
So, try to turn on the projector first, wait a few seconds, then boot the Pi.
If that works, generate an edit.dat file:
and put hdmi_edid_file=1 in your config.txt.
That way the projector recognizes the already running Pi, in my case.