I think L39 Nasty Plot is a big advantage for Slowking as its lack of Psychic for the E4 makes it advantageous to just set up and hit things with neutral Surfs/whatever. Much better than Slack Off for which you can compensate with Lemonades and whatnot. Nasty Plot is also not entirely impossible to be learnt on time for Clair.
I'd say that Slowking > Slowbro if one can afford to trade-evolve and Slowbro > Slowking otherwise (obviously). Physical bulk is nice for taking Hyper Beams, but Bruno can't hurt Slowking much either, and the special bulk is useful for facing the dragon-type specialists you meet. The 5 extra levels Slowbro gets aren't a huge advantage, as it will still go last in battle and both should OHKO most generic enemies with Surf (both meaning both instantly evolved Slowking and Slowbro from the wild).
I personally wouldn't ignore Slowpoke's possible early contributions in gyms 2-4 either. You're not taking Scyther's U-Turns definitely, but Slowpoke is one Pokemon who doesn't mind watching Kakuna/Metapod use Harden due to its special movepool - L11 Water Gun, L15 Confusion. Yawn can be used to help with Miltank, and it can use Confusion freely in Morty's gym on anything that's not Gengar - why would you be afraid of Lick anyway?
Finally, a physical Slowbro deserves a mention. You get Curse at the start, and Zen Headbutt is your physical alternative to Confusion (tbh I'd keep Confusion on special Slowbro/king anyway). Avalanche has 120 BP when taking a hit, and Slowdude will most certainly be taking those every turn. Curse up and Avalanche all of Clair's/Lance's Pokemon. You could keep Surf since Waterfall isn't a choice (and it doesn't require setup). Earthquake is there and helps with, like, Houndoom, but mostly redundant with Zen Headbutt covering poison-types better.