Kind of a Catch 22.
The Guest Network will not function correctly unless the Router Mode is set to DHCP and NAT.
But......enabling DHCP and NAT will mean that you will now have two routers both trying to control the network......the AirPort and the device that you call your "modem"......which is also a router. Not good.
Apple assumes that you have a simple modem, and you don't.
When you enable DHCP and NAT on the AirPort with the type of modem/router or gateway that you have, that will introduce a network error known as Double NAT, which can slow down your Internet connection and possibly not allow some web sites to connect. The AirPort will blink amber to complain if you try this, and a message will appear telling you that the AirPort needs to be in a Router Mode of "Off (Bridge Mode)" to correct the problem.
I would NOT recommend that you try to change the current correct settings on the AirPort to incorrect settings to possibly allow the Guest Network to function.
Since you have another router, why not use the WiFi from that device for your "guest" network?
Or, depending on the capabilities of your modemr/router or gateway, it might be possible to set it up to act only as a simple modem. Then......the AirPort could be configured in DHCP and NAT to allow the Guest Network to work correctly.
Not all modem/routers or gateways can be set up as a simple modem though. That would be a question for your provider if they furnished the "modem" to you.