Hey folks,
I finally got an answer from Apple Support on this. It was like, the fifth person I'd spoken to over the last few months. If anyone else has figured this out, I apologize, but I wanted to share it here.
The "messages" app saves a cached version of all attachments, sent and received, to your hard drive. This is independent of any photos or files you've saved to their respective folders - ie, if someone sends me a 1GB video of my band playing, and I save it to my photos, there is still a cached version of that 1GB video on my hard drive.
So, what I've done is go to Settings -> Storage -> Messages -> the little i with a circle around it -> and then deleted those files. I make sure that I've saved the ones I actually want to keep, but a lot of them are just kind of silly nonsense I've saved over the past few years.
As far as I can tell, this ONLY deletes the cached version on your hard drive, the attachment will still be there in the chat if it is one you received, but you should double-check that you've saved things to photos/files/wherever first in case I am wrong. I'm going through this very slow process - oddly enough, the amount of storage Messages is taking up according to the "Storage" app is fluctuating as I do this, but I think that's the fix.
Cheers.