In order to produce the Indicated Airspeed (IAS) our Airspeed Indicator measures the dynamic pressure, then produces an indication that is determined by the value of that dynamic pressure.
So if we are climbing at constant IAS then we must be climbing at constant dynamic pressure.
This means that as we climb 1/2Rho2 (TAS2)² must be equal to = 1/2Rho1 (TAS1)²
This in turn means that if we are to have constant lift to match our (assumed to be) constant weight, then our angle of attack must also be constant.
If we hit an upward vertical gust this will increase our angle of attack, which will in turn increase our lift and our load factor.
The increased load factor in the gust is proportional to the amount by which the gust increases the angle of attack.
The increase in angle of attack is determined by the relative magnitudes of the TAS and the vertical speed of the gust.
If we climb at constant IAS we will have a gradually increasing TAS.
If we assume that we meet a series of gusts of a constant vertical upward TAS while climbing at constant IAS, then in each case we will have a constant upward TAS (the gust) being added to an increasing horizontal TAS.
This means that the relative magnitudes of the Gust and the aircraft TAS will gradually decrease. This will cause the change in angle of attack to gradually decrease. This in turn will cause the gust load factor to decrease.
So the gradual decrease in gust load factor as altitude increases at constant IAS, is caused by the gradual reduction in the relative magnitudes of the (constant) upward TAS of the gust, and the (increasing) forward TAS of the aeroplane.
The only part in all of this that is played by the reducing air density is that this causes the aircraft TAS to increase in our constant IAS climb.
All of the above is of course a simplification of the situation, because the addition of the gust to the aeroplane TAS will also change the dynamic pressure acting on the aeroplane.