Bump Steer

These little cars respond to a wheel alignment. Poor alignment they can respond with poor behavior.
Can be tricky to diagnose problems as they only have 500 - 600 kgs. on front axle.
Tyres can make a huge effect on handling too along with wheel offset etc.
Are you sure it's bump steer you're experiencing ?
That's a shadowy looking bloke in the photo.

I've heard good reports on the RE003 so I doubt they are at fault. You describe classic symptoms of tramlining when you say "stuck in a wheel track and it pulls around".  RE003 are not inclined to do this. Some other tyres are fair buggers for this behavior. I personally avoid any tyre with anything resembling a sharp / square shoulder.
I will probably get mud chucked at me but I would suggest that front wheel off set and/or toe setting is more likely to be the culprit. (Wheel off set has nothing to do with wheel stagger). These cars originally came with +45mm offset.
McPherson strut design has bump steer in it but it rarely exhibits itself under normal driving conditions.
Those lowered King springs could be contributing to these symptoms too. There is very little bump travel available so on a one wheel bump it could very well be hitting right into the bump stop where loads immediately shoot to infinity.
What is commonly described as camber wear often has little to do with camber but is more likely to be as a result of incorrect toe setting for the camber being used.
All these opinions are purely my own observations and experiences over many years and are offered as a guide that I would hope is helpful. Soooo only reasonably clean mud please.

I'm with Falcon on wheel offset, magnified by wider tyres.
I get a different response in the steering wheel going over a parking lot speed hump at an angle.
First the outside edge of one front tyre strikes the bump first, then soon the inside edge of the other tyre - different amount of steering wheel torque. Would indicate offset as not ideal (given L&R caster, camber and toe are the same).
Are your wheels stock ? What are the tyre dimensions? I looked long and hard to find cheap aftermarkets that had offsets similar to stock. Settled for 17x7 on the front and 18x8 on the back, yes driving downhill.
 
Also what targa says is true, returning the car to original height will help also. 
If you lower a car, then all the geometry changes. There are compensating blocks that can return the control arm angles to original, similar for steering (tie rod) angles, but the factory dialed in the macpherson struts and wheels pretty well, perhaps a little soft. I've got Eibach springs and they are firm, gotta slow down over bumps or it is unpleasant, but with standard sway bars I have very little roll. It hooks around bends nicely.
The compensating blocks I mention help reduce body roll.