If, I had, for example, a Galarian Meowth and a Growlithe in the
Daycare, would it make regular Meowth eggs as well as Galarian
Meowth and Growlithe, similar to how Event Pokémon breed?
No, you would be getting only Galarian Meowth and Growlithe eggs.
So if you want all G. Meowth eggs, just breed with another G.
Meowth and you won't have to worry about another type of egg at
all. ^^