Lets see...
Falkner - Pidgey is a joke, Pidgeotto is slightly more threatening because of much higher stats but with few potions it's doable. Also, dunno how about others but i always accidentaly was overleveled for this gym, lv.9 for leader's ace is a joke.
Bugsy - Metapod and Kakuna are a joke, again. Scyther is not worth of trying, tho. Even if Beyleef can tank Fury Cutter or two, after few turns it will build-up too much power.
Whitney - Clefairy is a roulette for everything so meh. With its superior bulk, Reflect and poison powder Beyleef has a best chance to beat Miltank, tho.
Morty - Actualy totaly untouchable by Beyleef. Probably the worst match up for it. Immune to normal attacks and poison powder and resists razor leaf, pass.
Chuck - Primeape is meh, and Poliwrath is dead after one or two razor leafs. Bring confusion- and sleep-healing items, tho.
Jasmine - Magnemites can only strike your only with Sonic Boom, but on the other hand you have nothing strong against them. Steelix has pathetic Sp.DEF and Ground secondary typing. Iron Tail will hurt like hell, but doable.
Pryce - Seel and Dewgong have lowish stats and only Icy Wind, doable. Piloswine has pathetic Sp.ATK, but can fire off Blizzard and that's scary. Meganium could probably tank one Blizzard from Piloswine and cut it up with Razor Leaf, but player could probably have better options.
Clair - Dragonairs are meh, Her Kingdra normaly is a monster, but if player fire off Poison Powder onto it and stall it out then it's easy and boring.
I'm too lazy to analize E4 and Kanto, but overall E4 is a mixed bag for Meganium and Kanto has plenty of Grass-weakness.
Overall so feared opponent's for Meganium are... not that scary. Type match-up looks not promising and all, but then player realize that besides few exception they just lack fire power or proper movesets to beat Chikorita. Chikorita's problem is not that it "sucks", it just trades the fire power and Thunder/Ice Punch coverage for tremendous bulk and few support options, and players prefer offense over defense.