I'd rather not fill it in with a culvert because that would be cost prohibitive.

I think this is the first time I've
ever wanted to tell someone - "I wish your yard wasn't so big!"
What if you built a fake river using rocks? It would still let the water flow, and you could take advantage of those curves I see in the first picture. Something like this -
Now, the cost prohibitive part. You're going to have to go rock collecting. (Wish I knew what part of the country you live in!!) Where I live, it's mostly farmland. I drive 18 miles to work, and I pass four large piles of rocks that the farmers have removed from their fields. (Two of those piles are as big as my car!!) I have friends that have permission to walk some of the fields looking for arrowheads; I'm sure you could get permission to collect rocks! They
want rid of those rocks!!
You could look for free rocks on Craigslist or Freecycle. Or put out your own ad asking for free rocks!
Anytime you see new buildings going up, you can ask if they have rocks they want hauled away. (It might be beneficial to call construction companies and ask beforehand.) You could also place calls to whatever government department handles road construction.
