I have found an Armstead Carter, who signed a petition in Hanover County, Virginia in 1783 to ask those non-supportive of the American cause during the Revolution to leave the state. (It is also signed by a John Carter, but I cannot say that there is a relation.)
I suspect that this is the same Armstead Carter who served as a Private in Virginia's 1st Regiment between 1777 and 1779. I do not know that he applied for a pension.
I also suspect that this may be the same Armstead Carter who dies in Chesterfield, VA in 1795.