Hi, thank you for your question. Please confirm if you are married to each other and if the only issue is him getting a passport for the child if he was on the birth certificate?
Thanks for confirming. Firstly, as you are not married it is your decision whether he is included on the birth certificate or not. If he is not included and seeks to obtain parental responsibility which you do not agree to he will need to apply to court for a parental responsibility order.
In relation to contact, whether he has parental responsibility or not he is still entitled to seek contact and the child has a right to a relationship with the father which can only reasonably be restricted if there are welfare or child protection concerns - which will include the issues you have raised regarding him harming himself. If you were to restrict arrangements due to this there would be no detriment to you as you will be protecting your daughter.
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To add to the above
You need to be aware that in fact your ex can apply to the Court for a Declaration Of Parentage which would be granted (assuming he paid for the DNA test) and would allow his name to be added to the Birth certificate but which would NOT give him Parental Responsibility
To get Parental Responsibility he would need to persuade the Court to grant it to him - and your very serious fears of abduction should be sufficient to prevent that happening - unless at the same time the courts made an Order confirming that the child lives with you and that he cannot remove the child from your care
It is rarely wise to stop contact other than for a good reason - so offer contact at a local Contact centre for the time being