If one climbs with the same speed and the same weight as a similar plane would for level flight the angle of attack would be the same in both situations. Therefore I haven't included it in the equation.
And there's where you went wrong
Humour me - consider the problem with the aircraft deck angle as your frame of reference. Forget factoring L, D, T, and look at what happens to W...
weight on the wings reduces, so the AOA will reduce. Still works in your frame of reference, but you need to plug in real numbers to see how it comes to pass.