PJLink is based on HDBaseT protocol and Panasonic large projectors take Art-Net so you should be able to achieve what you are trying to do.
I'll be assuming you have a working understanding of Art-Net.
First, take a look at this
document on HDBaseT and PJLink, then take a look at the
manual for the Panasonic RZ12K (as an example) , go to page 164 for Art-Net Setup and page 197 for Using Art-Net Functions.
So, by using timed [Solid Color] clips, you should should be able to turns your projector on & off by sending Art-Net values to your projector on Channel 8.
Just make sure your Net/Subnet/Universe are set right.Also, if you have the right HDBaseT transmitter (Panasonic or third-party), you should be able to send Ethernet, Video & Audio (as well as RS232 and IR) on a CAT6 Ethernet cable.