I am trying to decide which option to go with. (or another if it is better) This is for a messaging type app where there will be high volume of notifications and database writes
One thing that I have recently learned as I am struggling to find a solution for it is that firebase does not offer any work around device to device notification ; while it does offer server to device push notification and it's pretty easy to set up. But the former lack of feature is very important and there's a conspiracy theory that it's because they are trying to push you to use other google products as well.
Or perhaps, since it wasn't developed at first they kept it the same. I have figured that app engine is a way to connect the firebase and the devices for this purpose and so I would lean towards combining both firebase and other google products in this case app engine. If you plan to do more back end processing such as image processing, etc, then you are looking at app engine and compute engine for sure which could be integrated with Firebase resulting in a hypothetically powerful backend solution.