Any idea how to download a small part of the chain for testing purposes?
As far as I know, there is no way to download arbitrary blocks (although I suspect it's not too hard to do if you implement your own client).

If you just want blocks for testing, you won't have much trouble finding bootstrap.dats that contain a few hundred thousand blocks. Alternatively, you could just run bitcoin core for half an hour or so and get a few thousand blocks from the start of the chain - The early blocks are mostly empty and download pretty quick.