One shade tolerant large shrub is Florida anise - Illicium floridanum is the name. If you planted 3 gallon ones, they would be 8 feet in about 3 years (most plants do the sleep/creep/leap growth thing). They like good moisture so they'd need to be watered well and mulched.

Other shade-tolerant ideas are Carolina Cherry Laurel and Hemlock. All these things are available in 3 gal and larger sizes at Buck Jones in Woodstock (on Hwy 140 going north from Crabapple about 8 miles). A non-native idea would be Aucuba japonica.

However, with such a narrow space and having to plant under oaks, I'd also consider non-planting solutions because sometimes plants just can't be the answer - they take time, especially in the shade. Could you build a pergola and train vines on it? Carolina jessamine is a fast growing evergreen vine with yellow flowers in the spring. I understand that you could not build a 15 ft fence, but you might be able to build a tall pergola/arbor. This would allow you stay inside a fixed width as well.

I have to agree with Esh. I have a line of oaks at the back of my property, and they can be hard to grow under. The heavy shade in some parts and then the dry dry soil. You also must realize that planting and and around existing tree roots and also putting ANYTHING in and around tree roots can really damage the trees over time. Last thing you need is to do all that work and then have tree problems in 10 years, in a narrow space..
The vine idea is excellent and could be done with a structure and some vines so easily. I'm thinking a few poles and some wire. Pretty soon you wouldn't be able to see the structure at all!