I'm up against a random Pokémon and I want to throw my master ball. Can it miss? If it misses do I lose it forever?
In Pokémon: Let's Go, Pikachu! and Let's Go, Eevee!, throwing a Master Ball will bypass the Poké Ball throwing mechanic, and will instead show a cutscene of the Master Ball catching the chosen Pokémon. If used on Legendary Pokémon in these games, they will attempt to smack the ball away, but will still be caught anyway.
Source
And, LGPE's Master Ball description states:
The best Poké Ball with the ultimate level of performance. With it, you will catch any wild Pokémon without fail.
Also, this video showcases different Master Ball animations in LGPE, and nah it doesn't miss.
Hope it helped!
No, the master ball cannot miss a Pokemon if you throw it in LGP/E. The game will do a special animation that is better seen than described. Watch this video Time stamp is 2:32:48