New York is very strict with their registration process so this can be a tricky situation. The best idea I have is find a friend that has a garage or place to store it.
If the only reason you would be adding your girlfriend to your policy is to save money, I wouldn't recommend it. At the end of the day, your insurance company would be insuring an additional person and vehicle so regardless of the multi-person or multi-vehicle discount you would receive, it probably wouldn't offset the additional insurable interests.
Coverage & Claims
Most insurance companies define their claims timeline as "as fast as possible." If you're handling the claim directly through your agent and that's whats holding up the process, you might find better luck with contacting your insurance companies claims department directly. You are able to file a claim independently of your insurance agent through the claims department.