Answer
No — per IFS R&D, Aurena doesn't support directly editing a Base Profile. The confirmed technical workaround is to create and configure a Personal Profile with the desired settings (like the saved search), then convert that personal profile into a Base Profile using Fndrr_Client_Profile_API.Make_Profile_Base__(profile_id) — the profile being converted can't be currently active (i.e., it can't be set as any user's default profile) at the time of conversion.