Okay, so forgive me if any of this is obvious, or already tried, but this is what I would do.
Personally I would simplify the network as much as is humanly possible, just to test the sync module. To do this I would borrow someones phone, and use it as a wireless hot spot. The beauty of this is that for the duration of the test, you can be sure that nothing else will be connected to that hot spot, and it will be the simplest possible type of network, with no complicated router settings, etc. When you create the hot spot, you can also give it an easy name and password, to aid in performing the tests. Don’t forget, it would only be temporary, so it wouldn’t matter.
Having created the hot spot, uninstall the app on your device, and then start from scratch. Download and install, then log in with your Blink account. Make sure you also reset the sync module with a straightened paper clip in the reset hole.
Choose add Blink device. Sync module should be the only option, because if your sync module isn’t currently shown, there can be no cameras, and you have to start with a sync module.
When it asks you which wireless network, choose the hot spot, on the other device, and continue with the setup.
If you are able to onboard it, your cameras should then magically appear. If they do, perform a couple of basic tests, and if they seem to be working, you can then move on to switching WiFi network.
Let us know if it works with a hot spot.