You can model the hole in your design for reference. If you are not actually cutting it you don’t really need to represent it though.
How are you going to use magnets to hold it down? Double sided tape or painters tape and CA glue would work if it doesn’t damage your record. If you are doing a lot of these you could create a jig from MDF which is square and has a circular pocket in the middle hold your record. Then you can always reference the lower left corner for X and Y. And maybe Z if your pocket is exactly as deep as the record is thick.
You don’t need to edit the gcode to send on/off instructions. Carbide Motion will either 1) tel you to turn it on/off or 2) do it for you if you bought a BitRunner to go with your machine.
Good luck. I haven’t seen anyone else mention PETG in the past but there are lots of posts on acrylic which I assume is harder.
Also, I should have pointed out that when using the diamond drag bit you don’t actually have the router turned on. It’s just there to hold the bit.
If you are using CC and CM and the stingray it seems to know not to turn the router on. Otherwise just make sure you don’t.