Well, here's a breakdown of your pokemon' types from what you've said:
Psychic/Flying
Water
Electric
Now the elite 4 types
Psychic (with grass, flying, ice, and water dual types)
Poison (with bug, flying, and steel dual types)
Fighting (with Onix (rock/ground))
Dark (with poison, grass, ghost, fire, and flying dual types)
Dragon (with water, flying, rock, and fire dual types)
Now, your three pokemon will do well against the following:
Lugia-grass, bug, fighting, poision
Feraligator-fire, rock, ground
Flaffy- water, flying.
This leaves psychic, dark, ice, steel, ghost, and dragon without something to counter against (not counting Feraligator's dark biting moves or any other non-STAB moves)
Therefore, you should consider investing in a good dark pokemon (Umbreon, Murkrow, Sneasle, Houndoom), an ice pokemon (Sneasle, Piloswine, etc) and a good fire type (Houndoom, Magmar, etc.) Also, evolve your Flaffy. Once you're utilizing all 6 slots in your party and they're all good to go, you'll have a MUCH better chance of getting through. Also, if you don't want one of the above choices, get a Dragonite. Dragon types are only weak to Ice and Dragon (although Dragonite is dual flying so that doesn't help) and you can teach it flamethrower with a TM, so it's a potential fire pokemon as well. Hope this helps!