Your spouse (on H-4) can apply for a SSN as soon as you have a EAD card in your hand. I do not believe that you need to give any reason to the Social Security administration as to why you are applying for a SSN.
Merely obtaining a SSN does not mean you have used your EAD. H-4 status is still maintained unless you use EAD to get a job or work in any way to get paid, after which H-4 status is lost and the individual is purely on "Pending AOS" status.
Even in this case, only the dependent has lost his/her H-4 status. The primary applicant still maintains H-1B status.
This is what my attorney has told me and I am in this situation right now where my wife is working and is using her EAD (after we obtained her SSN) and I am maintaining my H-1B.
Hope this helps