Bayesian Optimization (BO) has shown promise in tuning processor design parameters. However, standard BO does not support constraints involving categorical parameters such as types of branch predictors and division circuits. In addition, optimization time of BO grows with processor complexity, which becomes increasingly significant especially for FPGA-based soft processors. This paper introduces ASPO, an approach that leverages disjunctive form to enable BO to handle constraints involving categorical parameters. Unlike existing methods that directly apply standard BO, the proposed ASPO method, for the first time, customizes the mathematical mechanism of BO to address challenges faced by soft-processor designs on FPGAs. Specifically, ASPO supports categorical parameters using a novel customized BO covariance kernel. It also accelerates the design evaluation procedure by penalizing the BO acquisition function with potential evaluation time and by reusing FPGA synthesis checkpoints from previously evaluated configurations. ASPO targets three soft processors: RocketChip, BOOM, and EL2 VeeR. The approach is evaluated based on seven RISC-V benchmarks. Results show that ASPO can reduce execution time for the ``multiply'' benchmark on the BOOM processor by up to 35\% compared to the default configuration. Furthermore, it reduces design time for the BOOM processor by up to 74\% compared to Boomerang, a state-of-the-art hardware-oriented BO approach.