You said "spring rate"
The motion ratio is not going to be different, you'll just alter the starting point... but that doesn't matter other than slightly changing the loads on the end links. The anti roll effect will be unchanged as the load equalizes across, end to end... if you are concerned that the increased loading on the end links is going to cause them to fail, that's valid, but historically that is not what we have seen.
OK, I'm doing a poor job explaining on my phone, apparently.
Do you agree that for a given amount of suspension travel, the initial position of the sway bar arms will have an effect on how much rotation is induced to the main part of the bar? I hope so - because obviously there's even practical limits. You can't have suspension travel beyond 2x the arm length - the arms will be pointed straight up or straight down...
So given that, the relative amount of angular displacement for a given amount of vertical travel of the end of the arm is a function of the position(s) of the end of the arm(s). I agree, it's MORE complicated because the axle moves in another arc, the links themselves also relate the axle to the sway bar ends in an arc, and you have bushing compliance and other things, but to say that the starting position of the sway bar doesn't have an impact on the "effective linear spring rate" of the sway bar is incorrect.