Hey, welcome to the forum!
If it is fine in router mode but not in bridge mode then it could indicate you potentially need something like PPPoE, VLAN etc to be entered into the R2 but it could also mean the bridge mode isn't great on the router (that is the case with my setup).
I would suggest you take the Fritz out of Bridge mode and do the following:
Set static/reserved IP for the R2 on the Fritz
Enter the R2 WAN IP/one you just reserved into DMZ on the Fritz
Disable UPnP on Fritz
Disable WiFi on Fritz
Connect all devices to R2
This will then achieve the same effect as the bridge mode and allow the R2 to control everything etc.