What do you mean by reject? If LO is screaming, arching, stiffening, and seems to be in pain I'd investigate reflux and/or a milk protein intolerance. I can't think of anything more frustrating than trying to get a 2 month old baby to drink from a sippy cup.
WAY to early for a sippy cup. They need to be able to sit unassisted before you give them a sippy. Is she drooling a lot? My babies tend to start teething at 2 months.
Re: Rejecting breast and bottles