Marriage related

Yes you can register your marriage under Hindu marriage act.

1. As per Arya Marriage Validation Act, 1937, it is valid if both follow Hindu way of life and it is not valid if this "Muslim convert says she believes in both religions". 

2. Ideally both of you should have been married under Special Marriage Act and not under Arya Marriage Validation Act, 1937. It is the correct Statute for solemnization of marriage between people of different religions.

3. Now, privately ask the above quote /un-quote to your Muslim covert wife on her current religious faith. Also privately answer what does her Muslim father / her Muslim family say about this marriage that both of you have done privately.

4. Once you have her private reply, read with her Muslim family's current view on both of your marriage, the question of correct way of Registration of Marriage arises.