It is perfectly fine for mons to hunt their food in the wild, as it is for humans given the canon indicates humans used to hunt for their food as well in the Pokémon 'verse. For the most part, hunting for food is something that is expected to happen. I'd say you won't be disgusting anyone unless they purposely want to be offended, as many of us devour unborn chickens every day without giving a middle finger flips about it.
The problem, if any, is with depiction - most of us purposely don't want to think about the unborn chicken, or the fact that the meat we eat was once alive and had to be killed (and perhaps even born) with the express purpose to reach our table. Even less if those unborn chicken happens to be self-aware and sentient, not to mention overpowered in comparison to RL creatures, and could just as well have had an alternate path in life where instead of ending at your table he ends up as your neighbour's Starter Pokémon.
But you can not just ignore depiction and have a fanfic that reads as something reasonable and ntot just some wish-fulfilment. When you bring about a modernised frame into the picture, interspecies issues such as hunting become nontrivial. You say Castelia - a huge metropolis by the sea where many Trainers live or pass by and most likely a pole of commerce and trade. It more or less mirrors RealLife!NY, that should tell us enough.
That means that if you want humans, and probably mons, to use surrounding territory as hunting grounds, even if offscreen, they'll have to do it en masse, as they have a tremendous social engine to support. Even assuming the average mon does not need more than a single prey a day (more or less a given, as they can consume single berries then Hyper Beam everything around them to shreds) that means you have a presumed daily death toll in the same order of magnitude as the amount of (permanent, non-floating) population of active mons in the city. This is of course only a basis - add in some tech for syntethic food, etc, and it becomes easier - but also less trivial! (why would, say, a Lairon troop risk hunting in the wild when they can storm a Pokémart in the city outskirts for free food?).
For the purpose of acknowledgeing issues without explicitly trying to step on someone's toes or disgusting readers, I'd suggest that one of the first things you'll want to do is to acknowledge, in-universe, that the city's culture and society is adequately reshaped to reflect the changes you make in your setting. A city surrounded by hunting grounds means even a Trained mon can reasonably be fair game for breakfast, and that might be reflected in the city's civil law. At the same time, it might mean wild mons might follow some of the city's schedules and rites as part of their own activities (raiding and garbage collection being the two most obvious ones that come to mind right now).
So hunting is something that can perfectly happen, but if you want to actually acknowledge it without letting it become an issue, you'll probably have to be careful to scale your setting accordingly.
