The generate-exam embed flow has five new presets for trimming down the form your users see: hideSubject, hideStudentLevel, hideDuration, hideTaxonomy and hideQuestionTypes.
They pair nicely with the presets you were already sending. Set studentLevel and hideStudentLevel together, for example, and the level is applied without your users ever seeing the field — handy when your product already knows the answer.
All of them are listed in the exam generation flow docs.