Linux

This is honestly an extremely expensive (in terms of skills, maintenance, chance of messing up) solution for a small shop that doesn't mitigate at all the threats posed.

You said correctly, the employee has the final word on what happens to the data appearing on their screen. Especially in the case of client data (I.e., few and sensitive pieces of data), it might even be possible to take pictures of the screen (or type it manually) and all the time invested in (imperfect) solutions to restrict drives and network (essentially impossible unless you have a whitelist of IPs/URLs) goes out the window too.

To me it seems this problemi is simply approached from the wrong angle: once the data is on a machine you don't trust, it's gone. It's not just the employee, it's anybody who compromises that workstation or accesses it while left unlocked. The only approach to solving the issue OP is having is simply avoiding for the data to be stored on the machine in the first place, and making sure that the access is only for the data actually needed.

Data should be stored in the company-controlled infrastructure (be in cloud storage, object storage, a privileged-access workstation, etc.) and controls should be applied there (I.e., monitor for data transfers, network controls, etc.). This solves both the availability concerns (what if the laptop gets stolen, or breaks) and some of the security concerns. The employee will need to authenticate each time with a short-lived token to access the data, which means revoking access is also easy.

This still does not solve the fundamental problem: if the employee can see the data, they can take it. There is nothing that can be done about this, besides ensuring that the data is minimised and the employee has only access to what's strictly needed.