there's a section in the config page of the wireless modem allowing me to set the LAN range
Of course there is. Some user can have a valid reason to use a specific subnet. However, "I want" is not a reason.
Your "modem" seems to have at least three distinct functions.
* A modem. Modem converts one type of signal to another type. For example, between DSL and ethernet or between cable and ethernet. Modem does not route.
* A router. Router routes traffic between two (or more) ethernet subnets.
* Wireless access point. Kind of modem, converting between wired and wireless ethernet. AP does not route.
What is the easiest way to configure the network settings of all devices on a subnet? DHCP.
What is the usual host of the DHCP server? The router that routes all traffic between the subnet and everything else. The "gateway".
Lets assume that your "modem" does not route. A device at the ISP end is the router of your subnet and has a DHCP server. It hands out IP addresses for your devices. Public IP addresses. It probably has firewall rules to allow only the addresses that it has given. At least one address, but probably only one address. How many addresses? Should read in your contract.
My ISP is generous; five public addresses. Do I use them? Hell no. I have a router that takes one public address. My side of the router is different subnet. The router runs DHCP to configure all my devices. I do have a switch too; inside my subnet.
For my subnet, I have two options:
(1) A private subnet range and NAT at the router to hide it from the public internet.
(2) A public subnet range that is registered to me and the ISP knows to route traffic for it to my router.
Guess, which one I do use?
Practically every home user does use private subnet and NAT. All the consumer "routers" have that set by default. So do I.
However, I do use the other option too. The ISP supports IPv6 and offers a IPv6 subnet of public addresses. My router relays them to my devices. No NAT. No registering by me.
I bet that you don't "own" the 254 public IPv4 addresses that you have (implicitly) shown. Your "modem" probably does not route and therefore does not have DHCP, nor two IP addresses (one towards ISP, one on your subnet). Not routing, but passing traffic. A bridge, just like the switch.