Steve's Carts appears to harvest the Dynamic Trees 'trunks', but it does not check for them to be fully grown. As a result, it will harvest 'infant' trees, which provide no yields. Additionally, branches with a radius of 1-4 should drop sticks, and radius of 5-8 should drop wood (this is based on in-game testing; I have not verified anything from source code).
The leaves should drop acorns, seeds, cones, pods, etc based on the tree types. Carts appear to harvest the leaves and get these sometimes, but are unable to plant them, though they appear to follow the same placement rules as vanilla saplings.

I haven't looked into your source to see how you are handling the planting, but it may be as simple as adding some registryNames to a whitelist somewhere. As for the drops from the woods based on the radii, it would depend on how you manage the breaking of the tree segments in your code. Possibly could add a check to see if the wood has a "radius" NBT tag and modify it's drops accordingly?