Welcome to the downs,
Mellon!
Unfortunately, I'm not at all clear what your argument is, here– unless it's that "well, Bilbo isn't necessarily a reliable witness". Which may be so, but which is hardly a "case" in itself, and anyway cancels itself out, if you see what I mean.
Or is it that
"If
Arkenstone = marvellous jewel
And
Silmaril = marvellous jewel
Then
Arkenstone = Silmaril"?
–Which is simply re-stating the original case.
And if it's neither of those, then would you mind explaining again?