Some homebuilts fall outside the remit of the LAA in terms of gross weight, power etc.
These have the permit to fly approval and issue directly overseen by a CAA airworthiness inspector. They are relatively few, such as a turbo-prop Glasair.
I suspect that they won't be covered by this approval, but presumably the CAA could issue a similar approval. I suspect that such a process would be less straightforward and more costly than for a LAA type.