Are you using the preferred port order? These are indicated by dots on your hub.
Port 1 = .
Port 2 = ..
Port 3 - ...
It should be controller in port 3, keyboard in port 2, and mouse in port 1.
Also, power off your PS4 and make sure you power it on with the PS4's power button rather than the PS button on the controller.
As far as I know, there are no issues with the rival 310 and it does support onboard memory. With that being said, make sure that you have onboard memory enabled in the SteelSeries app on PC! You might be using 400 DPI and it is not enough to register a movement on screen. try pressing the DPI shift button to see if that makes any difference.
Edit: I just saw that you said you were pressing the fire button on the mouse with no response in game, that would eliminate the DPI theory. Try unplugging the XIM and plugging it back in while in the training range, does it flash green after a few seconds?