My understanding is that tyre pressure difference with and without car weight on the tyre is fairly insignificant, typically in the order of 0.5psi higher with car weight.
Wuss, you need to toughen up! I grew up in Sydney so know the road conditions. Used to drive around Sydney in a 'Nota Sportsman' (poor man's Lotus Super Seven). This car had almost rigid suspension due to light weight (500kg) and dampers that were always far too stiff. Just couldn't find dampers soft enough unless I spent a fortune on proper adjustable racing units!
This car was truly bone shaking, but boy did it handle!! (accelerated pretty damn hard too!). The problem with really excessive damper rates as on this car is that the car will grip and grip and grip, then let go big time with no warning whatsoever because you ran over a gum nut or something (embarrasing at the very least!).
Hmmm, could possibly be zeroing in on the problem(?). Are you running much neg camber?
I assume these tyres have larger tread blocks with narrow and / or few channels toward the outer edge and smaller blocks with wider and / or more channels (and sipes?) toward the inner edge? They are also on the softer side re rubber compound?
The smaller the tread blocks the more they'll tend to squirm around. The more channels (and sipes) the more the blocks will tend to squirm around. The wider the channels the more the blocks will tend to squirm around (because with wide channels the blocks can't as easily 'lean' against each other for support).
The softer the rubber compound the the more the blocks will tend to squirm around, adding to any problem that may be caused by a more open tread pattern (but also meaning that as a general principle a softer rubber will be a bit less responsive than a harder rubber, all else being equal).
These factors will subtract from tread stability and steering response because there will be a tendency to run at a higher slip angle, and probably more so with increasing neg camber as this will tend to place slightly more static load toward the inner parts of the tread (where block stiffness is less robust) and less load toward the outer parts of the tread where block stiffness is greater. This is less than ideal (with such a tread pattern) for initial response before any significant body roll has developed.
Things get better for the outer tyre as it gains pos camber with body roll in corners (i.e. slip angle is likely to decrease), but worse for the inner tyre. This is less bad for a double wishbone suspension with decent camber gain curve, worse for a Mac strut with a crap camber gain curve (and the camber curve for a Mac strut typically gets worse if the car is lowered).
At high lateral Gs most weight transfers to the outer tyres, so in this state its not all that bad (with the outside tyre at increased pos camber and more weight thus being carried on the stiffer outer part of the tread pattern), but from zero weight transfer up until a high weight transfer has been achieved its less than perfect. Keep in mind that a front drive car set up to handle well will tend to unload the inside rear a lot and the inside front not a lot (relatively speaking), so until we see quite substantial lateral Gs the inside front will still be carrying a substantial proportion of front weight.
Some types of assymetric tread patterns may be a reasonable compromise for a tyre used in the wet and dry, but not necessarily all that great in the dry. The front tyres on my car have fairly small tread blocks, but the channels are quite narrow so when a block distorts at least its able to lean up against another block(s) for some increased support. The rubber is also a bit on the hard side which means a bit less block squirm, which is good for steering response though of course isn't all that great for outright grip.
I don't mean to imply that all assymetric tyres are crap, just that an assynmetric tread is a compromise of which I'm not much of a fan in general principle.