Can Alice prove that she *did not* send an unsigned message to Bob?
There is no way to prove that the message is fake without some trusted authority (such as GMail's DKIM) that automatically provides some level of non-repudiation without Alice being able to influence it (Alice can't send a message with DKIM disabled if she's using GMail).
Alternatively, Alice can habitually sign every message and ensure that those who are receiving those messages know that any unsigned message is not to be trusted. This only works if her audience is technically-minded and not easily swayed by emotive language and unfounded claims. Since you say that the community trusts Bob more than Alice, this is really a no-go.
From a cryptographic point of view, there is no way to prove that a message is not authentic as long as the message and signature can be separated in any way whatsoever.