Well, much of that depends on what you're going 'inside' of. Rules change a lot. Some things that are inside, are considered to be outside, and would cause you to look a little odd for taking off your hat. Meetings where you're standing up, you're probally okay wearing the coat. Fancy dinner parties, the coat is going into a closet or Coat rack. There is also the speed of the meeting to factor in as well. If its a quick 15 minute stand up meeting, then you're okay not taking off your coat.
Mind you, I've not said anything along the lines of "The coat is totally acceptable in all these situations that it shouldn't be!" or anything silly like that. What I had said, was that if you're going into a situation where the coat isn't going to be acceptable to wear.. wear something else.
The only reason to wear the coat + suit, if they only provide the +1 to armor, is if you're going into a situation where the coat isn't going to be applicable all the time (but will be applicable at least some of the time), and you're locked into the Mortimer of London line of clothing for the situation you're going into.
Otherwise, wearing heritage + second Skin, or Sleeping tiger, or anything else, would be by far more advisable.