Any thoughts on whether an appeal lacking independent evidence would be seriously weakened would be appreciated...
Well, it depends which argument the evidence would relate to.
If you were to say that the current school refused to support your appeal, I think it likely a panel would take your word for it.
Proof that your son is underachieving could be a bonus, but I'm not convinced you need an 'independent statement'. Do school reports reveal anything about progress?
As Spider suggests, don't you have SATs levels to show his progression? It would be interesting to see what they were for English, maths and science in year 2, year 6, year 7 (and any other years!).
I would have thought the panel would be most interested in any evidence that supports your positive reasons for wanting a place at the school in question.
In the link I provided above, look how Dejavu proved her daughter's interest in science and her aspiration to become a vet. This was crucial to her successful appeal for a grammar school with a science specialism.