Well that simplifies matters, then doesn't it? You don't sign that birth certificate under any circumstances, period. You can always amend the child's birth certificate down the road but no freaking way would I do it without a paternity test. And support orders can absolutely be contingent on paternity tests, in fact, the court can order them (in which case, they're about twice as expensive).
BTW, says something if she doesn't want a DNA test. It's totally painless, you take something that is very much like a Q-Tip and rub it on the inside cheek of your mouth, another on the inside of the baby's mouth. Seal each in a vial thing, drop off at lab. That's it.