Is it permissible to postpone a votive Sunnah prayer until after the obligatory Fajr prayer if one wakes up a little before sunrise, fearing the loss of the obligatory prayer's time?

If there is a fear of the time for Fajr prayer expiring while performing the Sunnah (voluntary) prayer, then the obligatory prayer should be performed immediately. The Sunnah prayer can then be performed if there is still enough time; otherwise, it can be performed during the Duha (mid-morning) time. If the Sunnah prayer was vowed as a nadhr (oath), then this vow becomes binding and must be fulfilled, and this Sunnah becomes obligatory due to the vow.