I know it sounds crazy, and it might not even work in the context of Captain Toad or Smash, but I have an idea to get around Captain Toad's inability to jump without breaking canon or leaving Captain Toad with a major handicap.
In the Dragon Ball franchise, Mr. Satan/Hercule is infamous for his inability to fly or use any form of Ki. So, how do they balance that weakness out when he shows in games as a playable character? Well, given pretty much every single character can fly in Dragon Ball, they give Satan a jetpack so he's not put at a disadvantage.
So, I think we could do something for Captain Toad here. Rather than jumping naturally like the other characters, Captain Toad uses some kind of device that lets him simulate jumping. Of course, to balance out giving him a good ground game(which I agree with), make him like Little Mac with awkward jumps and weak aerials.
It lets him jump with forcing Sakurai to break canon to do so.