IDPA is a game trying to be reality.
That is not correct. IDPA is a game created around defensive principles. IDPA isn't tactical training and doesn't pretend to be. If you are not thinking of it as a game, then you might want to re-evaluate your attitude towards it since it tactically it teaches some seriously bad habits (although it is outstanding for honing your weapon mechanics).
Regarding the OP's question, if you are going to shoot IDPA more than once or twice a year, I would encourage you to choose the full-size M&P. The reason is that IDPA rewards you for going fast (although not at the price of accuracy). The full-size frame gives you more real estate to work with, easier reloads, more mass to counter recoil, and a longer sight radius. In the long-term, I think most people will find a full-size gun preferable for competition over a compact model.
If you are going to infrequently participate, then by all means use your M&Pc. They are great guns (my LTC gun is an M&P40c) and are fun in competition.
But, as with most things on the Internet, your mileage may vary. Try both and see which you like best. Then have fun regardless of what people like me write.