Hi Chris,
At this time, I'm not aware of a way to move Bitstreams between Items... other than at the database level.
Hypothetically, you should be able to move a Bitstream by changing the "bundle2bitstream" table in the database, as that table decides which Bundle each Bitstream belongs to. So, it may be possible to move them by moving them from one Item's "ORIGINAL" Bundle to another Item's "ORIGINAL" Bundle. (The "ORIGINAL" Bundle is just the default Bundle that DSpace uses to store bitstreams). If you move Bitstreams in this way, you will *have* to do a full reindex of the site (otherwise your search results will likely show the old location). I'd also recommend trying this out in a test/staging environment first, as I've never done this before... so, you want to make sure it works with no side effects.
Unfortunately, I do not believe this is possible from the REST API or UI at this time. (If I'm wrong about that, I hope someone corrects me!) The only way to achieve something similar from the UI would be to download the file from one Item and then upload it to a different Item, but that'd be a very manual process.
Tim