Very good thinking, sir. That appears to be exactly what was going on. Adding an entry for the Cisco device in /etc/hosts causes the endless DNS requests to stop. Thanks very much!
By the way, if anyone else has a similar issue, after adding the entry in /etc/hosts I also had to edit my /etc/syslog.conf to reflect the new hostname of the Cisco devices as opposed to the IP, so that syslogd would continue to accept syslog traffic from it.
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