What is the small amount of white, non-viscous, odorless fluid that was discharged during sleep, and what is its ruling after performing istinjāʾ, wuḍūʾ, and praying Fajr?

If you are unsure about the nature of the fluid (whether it is semen, wady, or something else), then ghusl (ritual bath) is not obligatory, because the default is that one's accountability is free from the obligation of ghusl until its cause is established with certainty. If you doubt the occurrence of an impurity, then build upon certainty, just as if you doubt the invalidation of wudu' (ablution), the default is that it has not been invalidated, or if you find wetness and doubt whether it is semen or not, ghusl is not obligatory due to doubt. And a woman who doubts about janabah (state of major ritual impurity) is not obligated to perform ghusl merely due to doubt.