Opportunity is missed by most people because it is dressed in overalls and looks like work. - Thomas Edison
You're 100% on the exact right track. Plug in a router with DHCP/routing off, use as an access point with the same network SSID, security and password. Aim it so the antennas work best towards your 3 season porch. If you are going for POE make sure that Ethernet has it from the switch it runs to.I recommend TP Link products for the best value for home use. They also have access point only devices to save more money. I use Unifi for work locations and my house, but that's not needed here.