Does the fur carry over to following days?

No it does not. I've skipped an entire day and did not get +200. You may have lucky finished or miscounted.

I started with zero furs, and now I have two bags and 39 furs leftover. That's a total of 439 furs gathered. With a knife that does not have any drop upgrades. If it was just lucky finishes, I'd need 100% luckies, with at least two being huge. That just doesn't work out. I suppose I happened to hit whatever the reset time for the counter is unknowingly about halfway.

Hmm...not sure then. You probably did shear near the reset time and it seemed like you had more.