2 Consoles Sharing The Same Ports. NAT Problem

Do you refer "bridged to the modem that the ISP provides" as a Wireless Distribution System (WDS)? Or is it actually your "modem" that is bridged?

What is the make and model of your modem?

That's usually the case , but, i have seen isp issued equipment do some strange things ...if you are renting their modem for 10 bucks a month like I was , the money you would save in one year would buy you a modem and a router.. .then if need be , you could
call netgear support for free since you have a new router but i doubt you would need to...

Your isp I'd probably charging you 10 bucks a month for that modem , I would go get my own equipment and give theirs back... saving money and solving problem....win, win..

I used to live with three other roommates in college and since I was the only one that couldn't tolerate anything but an Open NAT so we could all play MW3, I spent a lot of time on this very issue. We had four Xbox 360s with one connected to a older G Linksys
model. Anyway, I suggest looking into a firmware called DD-WRT. It's been about three years, but from what I remember the process with Port Forward, DMZ, etc on DD-WRT is relatively similar and I had success opening all four Xbox's NATs.

Does your router support Upnp? (Universal plug in and play) If yes, I would try just disabling the port forwarding rules for the time being and just enable Upnp. This should detect the required ports for whatever game/app you are using at that time and will
open them. (This is pretty unreliable however) but worth a shot.

Secondly, you said you have tried setting up putting your xbox in the DMZ but you still got a strict NAT. This should not happen if you set it up correctly. You need to make sure your Xbox one has a static IP which can be set-up on the console in network
settings. You should also reserve the Xbox One's IP in your routers settings.

Also remember a reboot of both your router and Xbox is required for any changes to take effect.