We've never had a problem crossing at the US border. Crossed a few times with our infants first time 3 months old. We just carried their birth certificate.
Consider yourself lucky that you already have a birth certificate. The first time we took DD to Disney, she wasn't quite 6 months old, and no birth certificate. I had to go to City Hall and get a "travel letter" basically stating that I had registered her birth and applied for a BC that had not yet arrived. You won't have any problems with BC and a HC to boot!