There are a couple of avenues to try. Certainly, an NDA is something you could respect, especially if you are being invited to "come see how we use your software". It sounds like your client is comfortable with it, even if you aren't. But direct observation would be my first choice.
You could set up a camera at a distance far enough from the screen to not resolve individual characters, and observe usage that way. You'll want to set the cursor to something large and high contrast for the test so the camera can pick it up. You could offer to let the client screen the video to assure themselves there is no confidential information visible before providing it to you.
Similarly, web analytic tools like CrazyEgg can track mouse movements without recording specifics of data. If it's a web application, it's a matter of adding a script call (and paying them some money.)
You can often record the screen flow by name, instead of capturing an entire screenshot or collecting the exact data being entered.
You can ask to place a camera on the screen facing the user (the sort built into every laptop these days.) Record the user's expressions during the test: concentration, puzzlement, exasperation, boredom, etc. Correlate the timing with the screens being viewed: how long did they stare at a screen prompting them to "Enter description of incident, using fewer than 40 upper case letters and no alliteration"?
I consider raw timing to be one of the best indicators of usability for a business product. If the users have to stop and think, it's wasting their time. Once I find the bottlenecks, I look at the screens: is the task too hard or complex? Is the wording of the prompt too confusing? Lengthy? Is it redundant? Cluttered? Stark? If the users have to re-enter the same invoice number on seven screens, it's a waste of their time. If the users are frequently making typos that force them back to re-enter the data, the data may be too hard to enter. You might not learn the specifics of the fault, but you can quickly locate it. And timing usually reveals nothing about the choices they made, the identity of the clients, or the nature of the data.
You can also consider setting up a usability lab, and provide your testers with sample sanitized data to be entered. Hold the test away from your customer's premises so as not to make the client uncomfortable that you'll see something they don't approve of.