Is there another way to setup SNMP?
Possibly, but it would require that you use an older Mac running the Leopard (10.5.x) or Snow Leopard (10.6.x) operating system, or a PC with AirPort Utility for Windows 5.6.1 installed on the device might work as well.
There are also methods that will attempt to install the older 5.6.1 utility on newer Mac operating systems, but this not officially supported by Apple, and it may or may not work. We cannot discuss those methods here, but even if it works, you will be required to lower some of the default security settings on your Mac.....which would not be recommended.

As you already note, AirPort Utility versions 6.3 and higher do not support SNMP at all.....even if the AirPort device might.
But, even if this works, it will only work for the AirPort Express, since the 802.11ac Apple AirPorts do not have firmware support for SNMP.
The problem is I cannot change the community password on the device to protect this data.
Not sure how that might affect things in your particular setup.