He objected to ____ sitting there.
I think strictly it should be in the possessive, but it is overly pedantic to do so.
strictly speaking it should be "me and Max". The test as to use of "and I" or "me and" is to remove the other person and see if me or I fits best and then use the corresponding construct.
Did no one learn anything from grade school. The only correct response is "Max and I". Although many would argue "Max and me" it just goes to show the degration of English brought forth by modren culture.