100% yes spring rate testing will help. Like many have said, cutting, grinding, prepping, etc will have an effect on the tire. I remember when working with Intercomp on their tire spring rate tester here is what I found out. Trust me we went through a lot of tires because of my connections back in the day.
I remember doing a tire test and we had 6 sets of tires, all cut and prepped the same, air pressure the same, rollouts were as close as they could possibly be. One set of tires was just plain no good. The kart was loose. Air pressure gain would always say more air, more air and that was confirmed by tire temps (on dirt, nay sayers I dont care, temps on dirt works you just have to know what you are doing). Anyway, about that time Intercomp sent me a tire spring rate tester to work with. I took all the sets of tires we used and measured their spring rate. All but the bad set were with in about 15 pounds of each other. Now this is speaking of the right side (RS) tires since they carry most of the corner load. The bad set had more than a 30 pound difference! We took that set of tires and put on the kart like before and as before it was loose. Then we swapped the RS tires and the kart pushed.
I put the bad tire on the spring rate tester and added air until the spring rate equaled the other tire. The kart then handles very well until the heat created air pressure build up. Then the handling went away later in the run. The reason, the tire had so much air in it that it crowned the tire pretty bad and it just gave up. Once again, tire temps proved it.
The real question is, is paying $1,000 for a tire spring rate tester worth it? For me it was an easy choice because Intercomp gave it to me.
The lesson is that you try to find matched tires as best as you can. If you do get a odd handling set of tires, swap the RS tires and see if you get the opposite result. If you do, get rid of the RS tires.
For the LR tire, the spring rate matters. If the tire has a softer spring rate than the tire before, then the kart will "sit" on the LR and the weight transfer will come more from the LF and less from the LR. This will change the handling of the kart but you can compensate for that with minor adjustments.
I always set my karts up with say 8 psi in all the tires. I never liked split air pressures but that is up to you.
Msquared