I want a Litwick (like really badly), but the only way I can think of to get it (without perfect weather luck) is by retaking the Fire Challenge. Is that even possible, and will Litwick still be there?
No. You cannot retake Gym Challenges. You can find Litwick in the Wild Area later on, however. You can also find it in the Motostoke Riverbank in intense sun and Bridge Field in heavy fog/intense sun, but judging from your description, you don't want anything involving weather.
Source (also this site)