Rainboe,
Been there, done that.
Initially I expected a something squared to jump out of the woodwork and bite...
I've now come over to the dark side
But....
JABBARA,
Can you give us a typical figure for the climb angle (or vertical speed and TAS) of your stabilized climb?
For instance, for a climb at 200 kts TAS and 2000 ft/min, the climb angle is about 6°. Since the cosine of 6° is about 0.995, the lift (normal to the flight path) at that sort of climb angle is reduced only by a very small amount (see all the earlier formulae...), and so is the AoA (we're talking a percent or so).
So unless you do some careful flying and 'measuring', you may not see the effect at all.
CJ