Are the tolerances of the optimum important in your case? (i.e., how sensitive it is to errors in the design parameters) If so, did you use any method to incorporate this information into the optimization?
It depends on the needs of the specific application. What typically happens is that you’d use BO to globally converge to within some tolerance and use the resulting surrogate to get a map of where the interesting regions are. You can then more densely sample these candidate regions or switch to a gradient-based method (via finite difference). For uncertainty information, we usually add this as a noise parameter, either on the input samples, or as part of the GP kernel.