I have a possible answer (possible because my installation crashed before I can test all of the features).
It is relying, as mentionned, on
biber as a backend.
In the preamble, you mention that you want to use biber as a back end (useful for tweaking the sorting), that you want to use the new sorting scheme you devised, and that you are going to play with the prefix in the bibliography.
At the end of the document, you are printing the different bibliographies with the help of keyword to differentiate between them.
I successfully managed to get the [F1], [F2] and then [B1] and [B2] behavior, but was not able to test the biber tweak due to crashing. However, I am fairly convinced that this is the way to go, due to previous experiments.
I am aware that this answer is not complete as it is not fully tested, but I wanted to launch it out so that people can play with it until I manage to repair my set-up (at work, where I have huge networking trouble, hence possible long duration before repair).