Unfortunately, the large body of research on neuropsychological processes of fear and how it works is not in accord with the usual gun folk rhetoric.
Very true. However the issue the OP's girlfriend seems to have is not spontaneous fear caused by a legitimate, imminent physical threat, but pervasive fear; with the gun itself as a stimulus. This fear exists outside the context of an actual imminent danger. If it is severe enough, it is considered pathological, and we call it a phobia. It is the kind of fear that can be allayed in her mind by putting her in control of the stimulus, if she is willing. Remember, it was the OP who identified her fear as "irrational," and, as far as we know, he is the only one here who knows her personally.
I would counsel the young lady to go elsewhere if the OP moves into macho ultimatum mode because he wants a gun.
That would be a fair course for her to take, although I don't get the sense the the OP would do any such thing. However, it does raise a question for those who councel him to abandon any thought of a breakup over this: If we deem it unreasonable for him to offer the "get okay with guns or I walk" ultimatum, why should we accept as a reasonable outcome her "get a gun and I walk" ultimatum? Because, if she does not come around to accept guns in her home, that is essentially what her answer may be.
This is where I think it fairly becomes a values test. If you see guns as merely a fun hobby with potential side benefits in the area of personal protection, her wishes will-and should-prevail. If you see guns not as sporting goods, but as essential tools of liberty*, than it wold be an evil act to allow anyone to deny you their use.
These all remain "ifs," though. We don't yet know what she will do. It is my hope that soon we get an update, and it is good news for our hero (the OP).
*Maybe if I had written "essential tools of a free citizen," it wouldn't have sounded like copy from a Libertarian pamphlet? Yeah, I guess it probably still would have.