Ok, I have a working solution here that provides me with most of the things I need over SNMP. Figured I'd share in case someone else comes across the same problem looking for a solution.
This requires that SNMP be installed with a working snmpd service running on your Pi-hole system. You also need the command line interface for SQLite 3.
On my Raspberry Pi 3 running Raspbian I just used the "Add/Remove Software" feature and installed "snmp-5.7.3", "snmpd-5.7.3", and "sqlite3-3.16.2".
On my Windows server I already have Net-SNMP.
Edit the file "/etc/snmp/snmpd.conf"
sudo nano /etc/snmp/snmpd.conf
...and add these entries:
extend pihole-permitted /usr/bin/sqlite3 -init /etc/pihole/init.sql /etc/pihole/pihole-FTL.db 'SELECT count(timestamp) FROM queries WHERE (status == 2 OR status == 3 OR status == 0)'
extend pihole-forwarded /usr/bin/sqlite3 -init /etc/pihole/init.sql /etc/pihole/pihole-FTL.db 'SELECT count(timestamp) FROM queries WHERE (status == 2)'
extend pihole-cached /usr/bin/sqlite3 -init /etc/pihole/init.sql /etc/pihole/pihole-FTL.db 'SELECT count(timestamp) FROM queries WHERE (status == 3)'
extend pihole-unknown /usr/bin/sqlite3 -init /etc/pihole/init.sql /etc/pihole/pihole-FTL.db 'SELECT count(timestamp) FROM queries WHERE (status == 0)'
extend pihole-piholed /usr/bin/sqlite3 -init /etc/pihole/init.sql /etc/pihole/pihole-FTL.db 'SELECT count(timestamp) FROM queries WHERE (status == 1 OR status == 4 OR status == 5)'
extend pihole-blacklisted /usr/bin/sqlite3 -init /etc/pihole/init.sql /etc/pihole/pihole-FTL.db 'SELECT count(timestamp) FROM queries WHERE (status == 4 OR status == 5)'
extend pihole-queries /usr/bin/sqlite3 -init /etc/pihole/init.sql /etc/pihole/pihole-FTL.db 'SELECT count(timestamp) FROM queries'
extend pihole-oldest /usr/bin/sqlite3 -init /etc/pihole/init.sql /etc/pihole/pihole-FTL.db 'SELECT MIN(timestamp) FROM queries'
Whenever you edit this conf file, you need to restart the snmpd service for the changes to take effect:
sudo service snmpd restart
You also need to create the file "/etc/pihole/init.sql"
sudo nano /etc/pihole/init.sql
...and add the following command to it to avoid "Error: database is locked" messages:
.timeout 10000
Example: to get the results from your Pi-hole system, you could do it in a manner like below. It retrieves number of permitted (forwarded+cached) queries:
snmpget -v 2c -c public 192.168.0.3 1.3.6.1.4.1.8072.1.3.2.3.1.1.16.112.105.104.111.108.101.45.112.101.114.109.105.116.116.101.100
Output will look something like this, and is easy to parse for monitoring software like PRTG:
NET-SNMP-AGENT-MIB::nsExtensions.2.3.1.1.16.112.105.104.111.108.101.45.112.101.114.109.105.116.116.101.100 = STRING: "18369"