Finny lives by certain ideals which, essentially, form a code of honor. Part of this code requires honesty. The opening chapters of the book present Phineas as being honest even about his trangressions.
His loyalty is part of his code too. Finny never breaks from his code, even if some aspects of it seem made up and arbitrary.
Finny refuses to lie about his height for the same reason he refuses to lie about anything else - because he is completely open and transparent and honest about his life and his activities. Finny is very comfortable with himself and sees no reason to make himself into something he is not.
Finny frequently looks at the world from a different perspective than anyone else, and what he sees and reports to others from that different angle may seem like a lie to those who don't understand his viewpoint. Finny refuses to accept that there actually is a war going on when he fears that he will not be able to become involved in a role that he would consider acceptable for himself. The "fat old men conspiracy theory" is his way of explaining how the fantasy of the war is being presented so as to convince the rest of the world that it is reality. However, it is not a lie - it is his truth, at that point in his life.