The Soldiers and Sailors System only listed the Charles Tompkins of the 1st Ohio Light Artillery without the middle initial. What was your source for that from your current post?
Artillery guys rarely take flags - unless they are bravely liberated from a home he passed by or from a surrendered unit. Also - where's the alleged New York connection?
The flag looks fine although a textile analysis would have been better to have for whoever bought it - but its provenance has more holes in it that Swiss cheese to me. A family that cannot recall what state or unit their ancestor served with? Simply strange. SOmething does not make sense here.