A way around digital signatures?
As owlstead wrote in the comments, you don't need a real public key infrastructure to use public key signatures for this. You can have A create a key-pair, then distribute the public key to B and C in advance, before any communications start.

A symmetric setup isn't much more complex either. You need two secrets, one between A and B, one between A and C. Calculate two MACs for each file, with either key. B can verify one, C the other. Even if E gets the secret key from C modifies a file, they can only calculate one of the MACs for modified messages.