I definitely wouldnt sign the certificate. But at the same time if there are no feelings,which there shouldnt be. You should have no problem being supportive and not stressing the woman into a miscarriage.
You need to put your foot down and begin to speak matter of factly. You need to start speaking in terms of "we" which will help disarm her. Let her know that "we are going to go to those doctors appointments". "We are going to take care of each other". Meaning that you dont move out until the baby is born. During this time you are to play the role of supportive father. Give her what she needs from support,massages etc. During this time you will use your phone to trick her into incriminating conversations. You will collect as many as you can. Getting her on record, in case this ever becomes a situation where you need to take her to court for your rights. Personally I dont condone using the system against another parent. But if she forces your hand you need to have some ammo of your own.