PHP-FPM was to blame every time. I even reconfigured the server to use OpenLSWS + LSPHP5, but this also did the same thing the moment we had more than a few users on the website. So I have moved us back to our old hosts for the time being, where the load average is consistently sitting between 4-5 - it also has access to 4 cores and 2GB RAM.
What is it that I am missing? The website is working and pages are being served to the clients in an acceptable time frame, but surely the CPU usage shouldn't be this high in order to serve ~122 users in a 15 minute time frame. I believe I have Zend Opcache and Memcached set up and running, but if that is the case and they are correctly configured, surely I shouldn't be maxing-out the CPU in this way?