I've encountered a Breeder that lets you fight her every time you go through Route 4 (Pokémon Breeder Irene, if I recall correctly). She seems to keep the same Pokémon and level every time you battle her though. Have you run into any other trainers like this, or is she the only one

All Breeders can be rebattled indefinitely. They'll also always catch you in their line of sight and force you to battle without changing or improving their team. It's not a glitch, as an NPC says...
"The Pokémon Breeder on Route 4 past this gate always challenges Trainers to battle when she sees them."
To continue on, only breeders will do these kinds of things. You can find special trainers in the Funfest Missions, though. Not to mention the regular amount of daily trainers you face from the last game.