Thank you both for the suggestions. HTTP 1.1 settings were the culprit.
Our Nagios instance takes the host name in the url parameters, so passing in the pool IP is not necessary (we will need to define a different monitor for each application). Checking the web application server will still be performed with an HTTP check directly to the app, but we will check other metrics via Nagios which is already monitoring and alerting for CPU, memory, etc. for other servers that the app depends on (DB, directory services, load on a specific SAN, network outage, etc). We will need to be careful defining what to do when BIGIP cannot contact Nagios.
In a sense, we will use Nagios as the external monitor, but will interface through the LTM's generic HTTP monitor to Nagios' web interface.
We are still on 9.2.5 due to some business constraints, but just purchased a new pair of LTMs which we will run V10. Will look into the external monitors as an option soon.