There are a lot of options here -and I mean a lot. A great many Pokémon can utilise weather effects one way or another - Swift Swim users in rain, for instance, like Kabutops, or Solar Powered Pokémon in strong sun, like Tropius. Abomasnow, Tyranitar, Hippowdon, Ninetales, Politoed - all of them have some direct connection to the weather as well.
You could broaden this a bit further by saying that most Grass-types can utilise the sun, if you think outside the box. Perhaps using Growth to increase leaf surface area could give a Grass-type sudden boosts in energy (which is, I admit, not totally realistic, but it's a nice thought), or something like that. The same goes for Fire-types, really, the sun drying out a battlefield and preparing it for a blaze. Or an Ice-type might freeze the rain to create icy spears. There are a great many possibilities there.
But - and it's a pretty big but - all of these Pokémon use only one kind of weather. Change it, and they lose their special something. Take your Sunflora out of the sunlight and put it in the hail and it no longer has much going for it (though you could argue for a way in which it might draw nourishment from the rain). If you want something like Castform - something that has a canonical ability to utilise all weather forms to support it in battle - there's a bit less choice.
You could start with Pokémon that can learn Weather Ball. These are: Lugia, Ho-oh, Castform, Roserade, Vanilluxe, Bellsprout, Snorunt, Celebi, Drifloon, Raikou and Bulbasaur. I'm pretty sure Lugia, Ho-oh and Celebi aren't quite what you're after, being somewhat legendary, so I'll discard them; Raikou and Bulbasaur only learn it by event, so you might want to discard them, too. That leaves you with at least five obvious candidates, excluding Castform, that all have a canonically-proven ability to use any kind of weather system to their advantage in some way. From that point, you're free to interpret that ability in any way you choose, as long as you can make the reader believe it's plausible.
That's not to say that's the only way to go about it, though. You could sift through the Pokémon that can learn the various weather-changing moves and see which of those can learn the greatest number of them, and which of them has any other defining characteristics that might make them suitable for what you want. Alternatively, if you were willing to confine yourself to one type of weather, then your options become a whole lot greater.
It really is mostly up to you. As long as you can convince the reader that there's a plausible reason for what you're doing, you can write whatever you like.
F.A.B.