Network bridging on Windows is a tricky thing compared to some other stuff if you are going to expect seamless. Have not had to play with it for a while (much less on Windows 10) but there will not typically be a "only connect when I say so" option in the bridge settings. Anything here is more likely the 360 riding roughshod or just the bridge being an afterthought (and arguably legacy) option for MS.
Simpler solution would probably be
Get a nice USB wifi adapter and stick that into your laptop. That way you have a secondary connection to your router and won't drop off whatever you are doing that wants more uptime. Still have to click around though.
Get some ethernet over powerline and go to the router itself, assuming that is an option for you.
I am curious though. If you plug the cable in after the hacked dash has booted do you still get the same problem?