I wouldn't use oak for that or PT if it's being painted.
A 6x6 cedar could be trimmed down.
A 4x4 PT can be wrapped with cedar boards to create a heavier post.
If you use 6x6 cedar you can cut the cost of the wood by using whgat I call a Cape Cod install (because that's where I learned it)
Pour a footing with a couple steel plates or angles that extend above ground a few feet and thru bolt the posts to these.