AirPort Extreme is disconnected from the internet when in bridge mode
This is not a real solution, but it might help if you have the same issue.

I still get the problem, and every time I get it, I restart the AirPort Extreme (by unplugging it from the outlet, waiting a bit, and plugging it back), and it works (until it stops working again). This happens around once a month or every other month, so it is not a big deal for me now.