If the customer is responsible for the network, all I would think you would need to do is just verify the camera is working. If you can direct connect or locally connect to the camera and do a continuous ping test (ping -t xxx.xxx.xxx.xxx) for at least 10 minustes or so without dropped packets (or I would say 1 or 2 packets in a 10 minute timeframe would be acceptable), and get live video from the camera by using the web interface on your laptop, and if you certify whatever jumpers or short cables you may have used connecting the camera to the customer's network, then you've discounted the camera being the problem.
But if you can't explain with confidence to the customer why your equipment is not the problem, then you may need to subcontract an experianced network technician who can confidently say the problem is not your equipment, but their network.
You may have properly used techniques and procedures to verify your equipment is working, but sometimes it's not just about doing the procedure, it's about having the confidence in your words that you know what is or isn't. And your'e really going to need an experianced and knowledgeable network person working on your staff if you are going to be attaching more security stuff on customers' networks, because IT staff will always blame and make you troubleshoot your stuff first so they don't have to do the work if they don't have to. That is just the way most of them are.