I would like the credit to go to Peter Woit for suggesting Section I.6 of Stephen Kudla's "Notes on the Local Theta Correspondence," which contains a nice proof, but he posted this only as a comment and the bounty ends today.
The text of Woit's comment is
There's a relatively straight-forward argument in Section I.6 of Stephen Kudla's "Notes on the Local Theta Correspondence", available at www.math.toronto.edu/~skudla/castle.pdf This may just be a restatement of the Rao argument. As mentioned elsewhere, it comes down to invoking non-triviality of the Hilbert symbol
