I installed a French Drain for a guy to drain his back yard, and setup was kind of like yours. The only out was downhill and the neighbor didn't want the water, and wouldn't let us ditchwitch across his lot to the development retention pond 
So I dug him a huge drywell 4' x 8'x 8', lined it with soil seperator fabric, dumped in 15 ton of limestone, sodded the top.
I do not know how your soil perks but this will hold a good amount of water in a rain, and will weep out the top if it fills up. You can let the weep run down hill.
Your other issue might be those trees. digging hole that big can disturb a lot of root mass.
So I dug him a huge drywell 4' x 8'x 8', lined it with soil seperator fabric, dumped in 15 ton of limestone, sodded the top.
I do not know how your soil perks but this will hold a good amount of water in a rain, and will weep out the top if it fills up. You can let the weep run down hill.
Your other issue might be those trees. digging hole that big can disturb a lot of root mass.
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