Any experts in using SOFTSUSY (the tool) here?
Well, even if you set parameters to 0, they can still run under the RG. Furthermore, the higgs mass is rather sensitive to a bunch of the couplings in the MSSM; I believe the program operates with the two-loop beta functions. So, just restricting yourself to a particular few parameters and asking how those affect the higgs mass is going to give you an incomplete story. If you're looking to actually physically remove particles and their couplings from the theory, that would require modifying the beta function, which I believe is hard-coded into softsusy. It might be modifiable; I don't know.