I was reading a message board post which stated that Joseph Arnold, the battalion's first field commander, held the rank of major. I have never found that rank associated with him. In fact, in post-war news articles, he is referred to as 'Captain' Arnold.
It may be that, in executing field command, both Arnold and Perry -- who each held the rank of Captain -- were granted some of the authority of the rank of major (like certain non-judicial punishment authority) because they were in command of four companies.
I actually experienced a situation like that in my own brief military career when a squadron which needed a field grade officer (major or above) to command it was temporarily left under the command of a captain. For that short period, special orders were cut granting the captain some of the authority, especially regarding disciplinary actions, of a field grade officer.
Perhaps that situation is what led to an association of Arnold with the rank of major.