Is there anyway to prevent backups from saving iMessage videos I send? Is there any way to prevent iOS backups from saving iMessage videos I send as part of a conversation?

I'd like to still backup all iMessage text, other iMessage content, and videos received, just not videos I send. I'm willing to use iTunes or any other software to do backups.

Why the heck would that be useful?

I send video clips all the time via iMessage and they account for roughly 99% of the back up space I use in iCloud storage. Yes backups are a good thing. The problem is many people, including me already use services to back up their videos like Google photos or something else.  So you effectively pay twice for the very large space taken by all iMessage videos. 

This is a relatively new problem, I'm guessing for many users, when iOS started offering the new (and sane) option to keep iMessages synced across devices.

Photos have the same problem, just to a smaller degree because they're orders of magnitude smaller.

Why not turn off backup of just iMessage content using options available?

Potentially years of conversations with friends/family, I think of as a small part of a life archive. 

Maybe 20 years from now, to read some insignificant exchange with a no longer living relative, I think is a cool option for the future. Time will tell if it actually ever has any value.

Why not just continue backing up videos elsewhere and then before a backup, delete them before starting a backup?

To start, I'm not sure it's possible for end users to specifically access and delete only videos that are attached as part of iMessage conversations. However even if it were, you might imagine how this creates a new manual process to worry about or a more complex process.

Environment?

iPhones and iPads. I try to ensure I waste enough money so all hardware is relatively current. Normally use latest iOS versions. Willing to use developer betas if it offered a new option for this.
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.