It seems we've now dealt with the original question, and we've all agreed lift and AoA decrease in a steady climb, even if only very little for small climb angles in a 'conventional' aircraft.
So we're now dealing with Rainboe's pet situation, a steady vertical climb.
Admittedly, at first sight we are just talking T=W+D+L, but since none of those vectors point in exactly the right direction, another diagram may be called for?
CJ