I’m willing to accept this theory about the World of Pokémon as canon. It just makes that much sense.
To be fair, not all plants are plant type pokemon, just like not all rocks are rock type pokemon. So vegetarians should be able to eat non-pokemon just fine.
But yeah, I don’t think any non-pokemon animals exist in the pokemon setting, so burgers are definitely pokemon.
Non-Pokemon are seen from time-to-time. Also, the Pokedex constantly refers to Pokemon that resemble animals as ‘snake Pokemon’ or ‘rat Pokemon’. This has some implications about the existence of non-Poke animals.
In the anime, there are normal fish and birds.
In other words, not all creatures are Pokemon.