If it were turned into a SSB style fight, there would need to be a few things that get addressed.
First, with league having the various forms of cc, those would need to fairly translate into a fighting game format. Since fighting games don't have cooldowns (95% of the time), Teemo would be able to make you miss all normal attacks and Soraka could make you unable to cast any special attacks.
Secondly, stealth would need to be adjusted in the name of fairness but no matter what will feel lacking. If there were to be non-cooperative multiplayer on the same screen then the person with stealth can either be visible and have that aspect almost entirely removed so that they can see their character, or have them be invisible and occasionally miss a jump or hit because they weren't entirely sure where they were by being fully invisible. Players would always see invisible traps from Shaco and Teemo be placed, and the only thing they could do is wait for them to expire. Let alone that this also would make the "smash ball" equivalent for someone like Teemo unappealing.
Items are added to the SSB games so that gameplay can be made diverse and add a bit of randomness that keeps every fight from being a boring punch-fest, especially at lower tier gameplay. If this game had items, it feels like it would be unfair to simply drop random occurrences of Chalice of Harmony or Ancient Coin as players may not want the item and be forced to choose whether they take it or their opponent that can use it does. If it still had the 6 item limit that league does, there would ideally be a way to dispose of your items one at a time so that you can dispose of weaker items without having to get rid of ones you like that you found.
There are a thousand other balancing decisions that would need to be done to keep things fair and fun on top of these.
TLDR: Some gaming styles don't transfer all too well from one style to another. League would need so many tweaks to it to fit that format that I don't believe they could still call themselves a league game, sort of like how Banjo Kazooie Nuts and Bolts shouldn't be considered a Banjo Kazooie game. You may have familiar characters and locations, but the gameplay is what everyone loves and that's what you'd change entirely.