Unfortunately, I don't have any numbers on false alarms, although they undoubtedly occur. None of the research I've seen collects data on this.
As for your dilemma, I would think very carefully before deciding to "walk in pushing." You would essentially be having an unassisted home labor, if not a home birth. If you feel you can work with your care provider and hospital and there are no deal breakers, then it makes sense to spend active labor, at least, in the hospital. If hospital and doctor policies are so unreasonable that you do not want to labor there, then you would be better off trying to find a midwife and planning home birth.
-- Henci