Fundamentally, once someone has some of the data, they have that data, and you can make no guarantees to remove it. The main question you need to ask is whether or not you're okay with limiting it to the data they've already seen, and what level of technical expertise they need to have to keep the data.

Making some assumptions for what's acceptable as a possibility, and how much you want to invest, I'd recommend having the data on a network-mapped share, and put a daily enforced quota for their access to it. Any data they accessed (presumably as part of their normal duties) is their's, and is "gone." But if you remove their access, they can't get any new data they didn't touch before, and if they were to try and hoover up all the data at some point to copy it off, they'd hit their quota and lose access for a bit (and potentially send you an alert as well). This wouldn't prevent them from slowly sucking out the data day after day.

If they only need to touch a small fraction of the customer data, and particularly if the sensitivity of the data goes down over time (data from a year ago is less sensitive than data from a day ago) this might be a decent solution. If they need to touch a large portion of the data, this isn't as useful.

Edit: another nice bit is that you could log on the network share (at your location) which of the customer data they're accessing and when. If you ever want to audit, and see them accessing things they don't need, you can take action.

I think the next best solution is the VDI one, where you run a compute at your location, and they have to remote into it. If they screen capture, they'll still save off whatever data they access, and if they have poor, or inconsistent, connection up your network it'll affect their ability to do their job (and depending how far away they are it might just be super annoying dealing with the lag). On top of that, it's dependent on how locked-down they need to be to do their job. If they need general Internet access, they could always attempt to upload the data somewhere else for them to pull it. If your corporate network has monitoring to catch that, you might be okay, but otherwise I think it's a lot of downside with a fairly easy way to circumvent.