Hi,
I am afraid there cannot be a clear cut answer to this issue, since there are various permutations involved in your business structure.
Because of the check-the-box regime in US, LLCs have an option to either act as a corporation (separate legal entity) or as a pass-through entity (considered same as the shareholder). The option which is chosen in US would make a difference in the tax treatment. If it is chosen to be a corporation, then the tax implication would be more since the after-tax money which you would receive from the LLC will be considered as dividend and you would additionally have to pay tax on such dividend. Again, the impact of choosing any option would also have a consequence on the Indian tax liability as well.
Further, if your turnover exceeds certain thresholds, then the LLC's POEM (Place Of Effective Management) may be considered to be in India since the LLC is managed from India.
A detailed study of facts of your case would be required to analyse the applicability of provisions of US & Indian tax laws and Double Taxation Avoidance Agreement (tax treaty) between India and US.
This is essential to correctly determine whether the income would be taxable in India or US or both and whether India will provide credit of US tax or US will provide credit of Indian tax.
I know that this would not resolve your query but I don't want to give you a wrong advice.