The P3P controller (GL300A) will always be a P3P controller with P3P firmware. It's identified as a P3P controller when connected to the DJI via internet through the app. I don't recommend messing with the firmware to gain compatibility with the P4, it may not be required.
I haven't used GL300A with my P4 yet, I use the GL658A Inspire RC with my P4. Here's what I would expect you to see with a GL300A linked to a P4. When you start the Go app with the GL300A
with the craft OFF, the app will show a P3P craft, because it's connected to a P3P controller. When the P4 is linked to that GL300A, when you turn on the P4 you will notice the app will see it and switch from P3P to the P4 portion of the GO app. You
may need "like vintage" apps and firmware to be compatible. IE, if you have old firmware in the GL300A, and brand new firmware in the P4, you might have problems. I'm not saying you will have problems with old and new mismatched firmware, but I've never tried that, and it wouldn't surprise me if there's issues. Especially if you have the new app with the new
P4P features (tripod mode, etc), using that with old craft firmware, you may have issues. I would never expect these compatibility issues to risk the flight integrity of the craft, so feel confident that isn't a risk. The incompatibility would be new app features in the app not working, or some other anomaly.
For starters, simply link the craft to the controller without changing any firmware. With props off, start up the craft motors at home and rev the motors like you're taking off. Make sure it appears to be OK with the camera linking and showing a video feed.
I do not recommend updating firmware unless you absolutely have to, as that may introduce other unwanted problems. Try linking only and see how that works. You can always relink the craft back to the GL300C if it doesn't work out. And as stated before, if for some reason you have to update the firmware in the craft, you'll need to relink the craft to the GL300C to accomplish that, as the GL300A will unlikely facilitate that task (that's what I have to so with my GL658A). But... as I mentioned, resist updating firmware to avoid the nanny state. On the other hand, if you have the most recent firmware already installed, and the most recent app, things should work fine.
There are ways to back-rev firmware, but I've never done it. You can google that if interested. DJI doesn't not support back rev of firmware, but hackers have figured it out and have an archive of firmware out there you can get.