No I wasn't picking on you. I too have had my difficulty's searching military. Particularly when 3 cousins have the same first and last name as their uncle and can't tell from the rolls which is the private, who is now Serg because someone else got shot, and who made it to Capt, at which conflict. And then they muster out again and the Serg is now a private and one is now Corporal, in different units!
I found the source below: http://wnyroots.tripod.com/index-1812.html
Which contains War or 1812 burials in a NY cemetery. Names of soldiers and next to it their commanding officer. Unfortunately the 'first' name of the officer is not given. Their are several Capt Mitchel's, many referencing Southern regiments, but if you scroll down to the very last one, there is an officer 'Capt Mitchel' listed as the officer in this company of some of the deceased soldiers, which seems more reasonable. ( It notes that these are not deaths from wounds but from other illnesses like typhus and pneumonia.
New York and Pennsylvania Volunteers
It might be worth seeing if you can find anything more about this group of volunteers and their officers.
Not really much help beyond where you already were, but googling:
New York and Pennsylvania Volunteers 1812:Mitchel
seems to be a place to start, as here's just one to start with:
MUSTER ROLL OF CAPTAIN MITCHELL'S COMPANY OF VOLUNTEERS BELONGING TO THE 5TH DETACHMENT PENNSYLVANIA MILITIA NOW IN THE SERVICE OF THE UNITED STATES AT BUFFALO, N. Y.
Capt. Andrew Mitchell
and below, maybe a less likely candidate from Maine.
The constitution and register of membership of the General Society of the War of 1812, to Dec. 1, 1895 .. (1895)
1891. Corliss, Augustus Whittemore, Captain U. S. Army, 55 Fort D. A. Russell, Wyoming. Grandson of Ebenezer Corliss, Private in Captain Daniel Mitchell's Company of (Detached) Maine Militia; 1814. Also served in North Yarmouth (Maine) Guards.
Maybe he had a pension which might help you in your search. Good Luck