Why not just use two USB dvd-rom drives? Or a desktop with two optical drives?
Data Hoarder
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I'd be surprised if you'd be legally allowed to handle such data with any of the programs suggested here. It'd be best if you reached out to whoever is responsible in your department.
Something like this?
Obviously this is make for ripping DVD/Blu-Rays, but combined with software like Automatic Ripping Machine, I feel confident that you could automate 90% of this. (Still would need to put disks in, and rename any folders that were created).
Based on the fact that you are dealing with medical files, you likely have HIPPA hoops to jump through, so using a unmanaged second computer might be out of the question. (As an it administrator I would not allow a raspberry pi or similar access to the VPN/Network). You may have luck asking IT for a second computer (that they control), and then do the USB drive option.