In this case from my previous experience with dome projection in general (same principles), I found out that there is mainly one way to do it depending the effect you want to add.
So the best way to achieve this is to have one more render pass as a final rendering, Blender wise to render twice your scene and add all the effects on the final rendering.
What I mean by that, DoF, Fog and other depth/camera position related effects will be done in the same stage where you rendering your pano (equirectangular map) and the rest pixel effects (such as bloom/glare, noise or any other seamless - texture like - effect is better to be done in the final stage or even better in real time (maybe BGE can help you on that). By rendering the sphere from a fixed point pin-hole camera (common camera) and adding your extra effects (pixel shader effects) you will avoid completely mangling with borders and margins.
An other way to do so is to use fish eye camera and render your whole scene as a disc and then projected again it on an hemisphere (>360 degrees), it is not perfect but it will also do the job if you dont really care about 100% accuracy and having a perfect sphere and you wont have to split the rendering procedure in two stages to achieve this.