Putting father’s name on the birth certificate

I think morally and to be fair to the child and father, you should put the father's name in the child's birth certificate. In addition, there will be inheritance issue in future. Just my two cents worth.

U can call the ICA to check. If you are lawfully married to the current husband usually will be automatically to them.

Unless you're unwed but in your case, married but intending to separate

I know someone who did provided the father don't sue you