The book 'Lunenburgh, or the Old Eastern District' By J. F. Pringle (1890) lists "Alexander Grant of Charlottenb'g" as among the "officers, non-commissioned officers and men of the R. R. N. Y.- King's Royal Regiment of New York (Sir John Johnson's Corps.) who settled in Upper Canada."
This book cited an earlier book "The Centennial of the Settlement of Upper Canada" by Hunter, Rose & Co as the source for the Regimental Roster.
The Roster denotes individuals that joined after the Regiment was formed in 1776/7. Alexander Grant was not listed as a late arrival, so I can only assume that he joined in 1776/7.

There is an Alexander Grant listed on the "RENT ROLL OF THE KINGSBOROUGH PATENT" as beginning to pay rent to Sir John on 25 Sep 1780 for 100 acres. The Rent Roll was the list of the original renters in TRYON County, New York State, was
from a submission Sir John Johnson made in a Loyalist Claim for losses. This Rent Roll was found in the Public Record Office in London, England. Many of the people listed served in the King's Royal Regiment of New York.