Sibling of #65254 (main-slot endpoint preservation): the auxiliary scope of
POST /api/model/set dropped the request's base_url/api_key on the floor, so
an aux slot pinned to a custom/local endpoint silently depended on
model.base_url — and broke the moment the main slot switched away and
cleared it. The aux resolver already reads auxiliary.<task>.base_url/api_key
(_resolve_task_provider_model); this persists them.
Desktop side: setAuxiliaryToMain / applyAuxiliaryDraft now carry the
user-defined provider's api_url as base_url, mirroring applyMainModel.
