Wanted to hop on the train of the Four kings au with some head canons! Going off of the 2023 Mario movie and just what I know about Bowser in my experience (which isn't a whole lot so I'm sorry if some things are not accurate)--
Much like Bowser, King Raph would be a powerful and often feared ruler to the other Kingdoms. He is seen as aggressive and impulsive, quick to react with a fight if anything offends the King. His anger can blind him, which as much as it can get him what he wants- it also can be his downfall.
But he is not heartless. Raph takes good care of his kingdom and his people, which leads to his citizens being very loyal to his cause. So long as you follow his rule, you are protected under his care. I mean, have you seen how the King fawns and babies the stray animals he finds?
Big puppy energy. This man is very lonely and kind of awkward when it comes to anything outside of war and conflict. He leans more towards a tsundere way of dealing with his feelings, brushing them off as quickly as they come with a huff of smoke from his nostrils or a growl. He is the King. He must not show weakness. But around you, he like an excited puppy. Stumbling over his words and twiddling his claws like a love drunken fool.
Would do absolutely anything for you. Want to sit on his throne? Go ahead! A piranha plant bit you? It can't bite you anymore when its ash! He would conquer any land or defeat any foe if it meant keeping you by his side. Just say the words and he's sending armies to take care of it.
Has absolutely no idea how to be normal about crushes. Raph always just takes what he wants, why would you be any different? Expect to be kidnapped a lot by this big guy. Doesn't matter if you live in his kingdom or not, he will always find a way to snag you away to spend time with you. Though he is very gentle with you, so you are never harmed in these schemes. You seem to be the only person he can't be truly angry at. It's... a bit unnerving to his guards sometimes. But if keeping you around means they get thrown into the dungeon less, they don't really mind.
If you choose to come with him willingly, he is over the moon! He is all over you giving you affection, carrying you around in his arms like the princess he sees you as (Though he hopes to soon make you a Queen). Despite all his spikes and his horns and sharp teeth, he is very cuddly. He'll often rest his head in your lap with his arms wrapped around you, churring loudly as you give him head rubs and gentle scritches. His tail will thump loudly behind him as well, which has knocked things off of shelves and tables more than once with the mini earthquakes it produces.
Is very quick to jealousy. King Raph is often seen as scary or seen as a monster most times, which leaves him insecure about his appearance from time to time. He's afraid that you're scared of him too and will leave him for someone less 'monstrous' like his brothers, making him very clingy each time he thinks you're showing any signs of disinterest towards him. You're the one person he does not want scared of him. Please don't leave him or see him as a monster. His heart wouldn't be able to take it. :(
Overall, Raph is very brash and impulsive, but very kind and compassionate as well. Gives kind of wet cat energy in a weird way. He really is just trying his best here, even if his actions come off as intense.