Is there anyway to prevent backups from saving iMessage videos I send?
In your original question you mention iCloud backups, so I'll address that first. If you don't mind not backing up Messages at all, then you could turn off Messages in Settings app > Your Name/Apple ID > iCloud > Manage Storage > Backups > This iPhone under "Choose Data to Back Up" - that is assuming 1) you don't use "Messages in iCloud" and 2) you don't mind not backing up the entirety of Messages including just the message text themselves.

There's no such customization for making the iTunes backups that I know of. I make some software for filtering an existing iTunes backup before restoring, but that won't help with the storage on your hard drive.