When I am faced with a similar situation, I use either a Windows 2000 or Linux live CD in a virtual machine and set it to bridged mode. I set the IP to the same as the interface on the router and access the web interface that way.
The other was is simply to manually change my IP to match that of the router - as you shouldn't need to change settings to often, I do not really see this as that much hard work.
Lastly, unless you have a reason for setting it like this, you may want to enable NAT on the modem and use it as a full router for your network.
How do I access my modem's GUI when it's in bridged mode?
The few i've used, when in bridge mode, have no web interface or IP. (maybe yours does)
With those, you have to reset it(holding the reset pin) to get to the web interface and that takes it out of bridge mode. With yours, there may be other ways, but You can reset it too like that and get out of bridge mode and to a web interface.
no way to access the modem GUI if I want to do things like turn off
its wireless antenna. (I was also thinking about disabling its NAT and
DHCP functions, although are those are even relevant when it's in
bridged mode?
With ones i've used, NAT and DHCP definitely aren't done when it's in bridge mode, I can't comment on others. I'd be surprised if it did wireless but you can check just to make 100% sure.. If you know the SSID (which you'd need to know to know how to connect to your own router when it's in regular mode), then look for that SSID to see if it's broadcasting. I'm sure you'll find that's off too, in any of these devices when in bridge mode. I suppose you could also prove that it's not doing NAT, with the tracert command tracert www.google.com and look for private IP addresses like 192.168 or 10.0 and you should only see one of them, if double NAT I suppose you'd see two.
Note
I have edited my answer, my previous answer said "When it's in bridge mode it has no web interface, no IP." I removed that since it's false for DDWRT and possibly false for some makes/models. The other answer says no , however that's not true either. I know this to be the case(the web interface turning off when in bridging mode), with some router/modem/switches, specifically with at least some, belkin ones e.g. http://forums.whirlpool.net.au/archive/869904 as mentioned here "if the belkin is correctly set to bridge mode you can not enter its gui web interface , as it is just a modem". It is true of DDWRT that you can still access the web interface.. Though quite likely that his particular make/model can, as the other answer claims to have used similar models to the OP, and says you can, i.e. you can with his. Though the other answer then talks about DDWRT which the OP may not be running. And the other answer doesn't specify what is going on with the OP's device if he has no access to his web interface.
Best Answer
Your first graphic properly depicts the modem in bridged mode. The second graphic does not correctly depict bridged mode. It's actually not a valid network scenario since a device cannot have the same IP address assigned to multiple interfaces.
When your ISP modem is not bridged it acts like a router (i.e. an OSI layer 3 device). The public IP address associated with your Internet connection is assigned to the modem's WAN interface. Further, its LAN interface will likely have a private IP address (e.g. 192.168.1.1). This means that if you connect your own router to the modem, its WAN interface misty be assigned an IP address on the same subnetwork as the modem's LAN interface, for example 192.168.1.2. Then your router's LAN interface would need to be configured to use a different subnetwork for its LAN interface, e.g. 192.168.10.1
When your ISP modem is bridged it stops acting as a Layer 3 device and instead simply forward all of the frames it receives between your router and the ISP's network*. As a result it doesn't have any IP address associated with it and you must use your own router (using your own router is optional if using the modem in non-bridged/router mode). The public IP address for your Internet connection is then assigned to the WAN interface of your router and you're free to use whatever IP address for its LAN interface.
You don't say which device's configuration page you reach when you connect to 192.168.1.1, but I'm guessing it's your router's. If so that's actually expected behavior as explained by the above description of bridged mode.
*Even in bridged mode the modem will respond to a connection attempt to the management IP address configured on its LAN interface. This is necessary so you can still access it to change its configuration.