Recent theoretical studies of topologically nontrivial electronic states in Kondo insulators have pointed to the importance of spin-orbit coupling (SOC) for stabilizing these states. However, systematic experimental studies that tune the SOC parameter $lambda_{rm{SOC}}$ in Kondo insulators remain elusive. The main reason is that variations of (chemical) pressure or doping strongly influence the Kondo coupling $J_{text{K}}$ and the chemical potential $mu$ -- both essential parameters determining the ground state of the material -- and thus possible $lambda_{rm{SOC}}$ tuning effects have remained unnoticed. Here we present the successful growth of the substitution series Ce$_3$Bi$_4$(Pt$_{1-x}$Pd$_x$)$_3$ ($0 le x le 1$) of the archetypal (noncentrosymmetric) Kondo insulator Ce$_3$Bi$_4$Pt$_3$. The Pt-Pd substitution is isostructural, isoelectronic, and isosize, and therefore likely to leave $J_{text{K}}$ and $mu$ essentially unchanged. By contrast, the large mass difference between the $5d$ element Pt and the $4d$ element Pd leads to a large difference in $lambda_{rm{SOC}}$, which thus is the dominating tuning parameter in the series. Surprisingly, with increasing $x$ (decreasing $lambda_{rm{SOC}}$), we observe a Kondo insulator to semimetal transition, demonstrating an unprecedented drastic influence of the SOC. The fully substituted end compound Ce$_3$Bi$_4$Pd$_3$ shows thermodynamic signatures of a recently predicted Weyl-Kondo semimetal.
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