Verify file authenticity from Flash client without revealing key
Is there another way that I can verify a file's signature on the client side, without exposing the method used to create that signature?

Public key crypto. You have only a public key at the client end, and require the private key on the server side to generate a signature for it to verify.
What is the attack you're trying to prevent? If you are concerned about a man-in-the-middle attack between an innocent user and your server, the sensible choice would be TLS (HTTPS). This is a pre-cooked, known-good implementation including public key cryptography. It's far preferable to rolling your own crypto, which is very easy to get wrong.
I'm building a Flash application to run on the web, where users can visit and create their own content in conjunction with my service (built with Python). Specifically: the user sends in some data; some transformation is performed on the server; then the finished content is sent back to the user, where it is rendered by the client app.
I want to be able to prevent the client from rendering bogus content, which I can do by passing a keyed hash along with the main content, generated by the server. The client would then use the same key to hash the content once again, and confirm that the hashes/signatures match. If there's a mismatch, it can be assumed that the content is inauthentic.
The problem I have is that keeping the key inside the SWF is insecure. I've considered a number of ways to obfuscate the key, but am learning that if an attacker wants it, they can get it quite easily. Once an attacker has that, they can start creating their own content to be unknowingly accepted by the client.
Is there another way that I can verify a file's signature on the client side, without exposing the method used to create that signature?