The widespread adoption of bulk metallic glasses (BMGs) in aerospace and biomedical industries requires topology-optimized architectures that conventional manufacturing cannot achieve. In response, BMGs have been investigated for powder bed fusion (PBF), but the process remains challenging due to narrow thermal windows, expensive feedstock, and limited data. This study introduces a constrained multi-objective Bayesian optimization framework to optimize key PBF printing parameters, including laser power and scan speed, to maximize hardness while preserving the amorphous state of the printed BMG. Hardness is optimized as the primary objective with density incorporated in the scalarization to regularize the search space, and amorphous retention is enforced through a feasibility probability learned by a logistic classifier. Surrogate models are compared, including Gaussian process, Bayesian additive regression trees, Bayesian multivariate adaptive regression splines (BMARS), and a Bayesian attention neural network. Acquisition scores are computed with constrained expected improvement and are maximized on a uniform grid over power and velocity. Superior predictive accuracy is obtained with BMARS, and 95% credible intervals are calibrated to the measurements. A high-hardness region at high power and low velocity is localized by the surrogates. A fully amorphous sample at 60 W and 1300 mm/s is produced, and a hardness of 1010.4 HV is measured in agreement with the predicted high-hardness band. In conclusion, the study establishes a data-efficient process-window discovery method for BMG PBF, produces an interpretable process map, and demonstrates a screening framework suitable for constrained experimental budgets.