While I agree that you can't insert a (non-free) action into the middle of another action, I think it would be perfectly reasonable (and well within DMG guidelines) to instead charge extra squares of movement for passing through a closed doorway (which would afterwards be considered open).
How I'd rule it if it came up in my campaign is that a regular, reasonably maintained, easily opened door would cost an extra square of movement (assuming the character had a hand free to manipulate the door, if it requires such). A stiffer door, or one that swings towards you instead of away, might cost two extra squares. If a door is so difficult to open that it's worth three or more extra squares of movement to go through, then should it really be a minor action to open when adjacent?
After the character passed through, most doors would be considered to remain open. If they want to close it after passing through, then I'd probably require them to stop and spend the minor action.