I haven't had your problem, but I have a similar setup. So, maybe my observations can help you with where to start looking for your problem.
I have a P2, GoPro Hero3, Zemeus 2-axis gimbal, iOSD Mini, ImmersionRC downlink transmitter, Black Pearl 7-channel monitor.
With my P2 powered down, my Black Pearl has a screen full of static.
With the P2 powered up, the screen goes black with the iOSD mini telemetry overlaid on the black.
With the Hero3 powered up, the camera image appears on the screen with the telemetry overlaid.
You say that your screen goes "blank." I assume that this means that it goes "black", not static. Let's assume you mean "black."
This would indicate to me that your downlink is still functioning. Otherwise, you'd get "static" on the monitor. It's got to be either the GoPro or the connections between it and the downlink.
You say that if you cycle the GoPro power, the image returns for a while. This seems to indicate it's either the GoPro itself, or perhaps the connection to the main board. Hopefully, not the GoPro itself. But in pushing the GoPro power button, you're probably wiggling the connector on the gimbal and/or the connector strip between the gimbal plate and the camera. Make sure these are all seated firmly. The connector on the wires running to the main board inside the bird is in a clumsy, inconvenient place for big fingers to insert and seat it. Check this one. Also wiggle the wires, there may be a break in a lead.
My blue-sky guess is that you've got a mechanical problem....That is, a loose connector or a broken wire, which when wiggled a little makes contact for a while.
Other than carefully checking all those connections, I have no other suggestions. If it is the GoPro itself, you may have a problem beyond fixing yourself. I know a couple of guys with GoPro's, and the next thing I'd probably do is borrow one of them and see if the problem follows the camera.
If your screen actually goes to "static" (instead of "black" as I assumed in the discussion), you'd also have to suspect your downlink transmitter and the wiring to it. But I don't know how recycling your GoPro power would affect things that far downstream.
Good luck.
