Hi,
In the given facts, since the entire business is effectively operated and managed by you from India, the place of effective management ('POEM') of the business is in India. Thus, the profits of your business will also be taxed in India. It may also get taxed in USA because LLC is registered in USA. (PS - while poem guidlelines are applicable only when sales exceeds 50 cr.,but income tax department will certainly try to catchhold this under other provisions such as GAAR)
Yes, you will be able to get tax credit of the taxes paid in USA while filing Indian tax return.
however, you may undertake certain planning to come out of this situation and your profits will be taxed only in USA.