Are there any non-surgical methods of dealing with imperforate anus?

Yes. Some forms of imperforate anus can be treated with dilation or stretching.
No. Imperforate anus is a surgical emergency and requires immediate surgical evaluation, preferably by a pediatric surgeon experienced in the procedure.
No. If it is true that there is no anus the child will need surgery. The extent of surgery and the steps involved should be reviewed with a pediatric surgeon.