Obviously putting the test data under VCS wouldn't work
Why not? And what about LFS?
Even without LFS, you need to think twice before discarding version control solution. You tell only that there are some files which have a size of a few megabytes. No further indication.
A dozen 5 MB files which change once per month for a 12-months project is not a big deal: any version control will handle that painlessly.
Hundreds of 5 MB files changed nearly daily for a 2-year project could bring some version control systems/servers/networks down and require additional preparation.
If you're in the first case, simply create a dedicated repository using your preferred version control system.
If you're in the second case, then you need to be very specific: how much files? How frequently are they changed? What is the exact average size? How long is the project? From there, you may either use an existent solution, such as LFS, or create your own, where files will actually be stored on a NAS, but never accessed directly, but only through the custom version control.
If you're in Microsoft world, FILESTREAM feature of Microsoft SQL Server could help abstracting away the actual storage of the files. Eventually (although I don't know Microsoft SQL Server well enough), you might be able to combine it with CDC to track the actual changes. Also, SharePoint provides you with the capability of versioning binary files while storing them efficiently if only part of a binary file is actually changed.