If I went the center channel route, I would try my best to do it right. However to fit the mantel would probably require small drivers (maybe 6") in a horizontal MTM configuration. Then maybe use either identical drivers for the L/R speakers or perhaps upsize the drivers from same manufacturer. Maybe this is what you mean by doing things the wrong way!
Here is a picture of how I might integrate 3 pi speakers. Distance between speakers is just over 15' and distance to listener is about 12' so that's good. TV is probably 55" so good for a 12' viewing position. Cost of the 3 pi kit is actually under budget. Speaker front baffles are about 14" so not too visually dominant (would that dimension work?). How about the stub wall (pocket doors) - is 34" enough to make the corner horn work correctly?
Actually, all of your base kits are within budget. The 4 and 7 pi seem too large with the 15" drivers. What speaker would you recommend and what width would the baffle have to be?