In SS, the wild Lugia is level 45, so it should know extrasensory, rain dance, hydro pump, and aeroblast.
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It's not immune to false swipe or sleep and doesn't know any annoying moves that damage itself, heal itself, or leave the battle. You should be able to catch it using false swipe, a sleep move, Poke balls, and not much else.
The best Pokemon that learn false swipe is Heracross. Unfortunately, Heracross can know false swipe only from breeding. It's a lot easier to use either Scyther or Pinsir, which learn false swipe by TM and are almost as strong as Heracross.
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Scyther @ sharp beak
Ability: technician
EVs: 100 Atk
- wing attack
- swords dance
- false swipe
- X-scissor
Pinsir @ muscle band
Ability: hyper cutter
EVs: 100 Spe
- X-scissor
- swords dance
- false swipe
- return
(Yes I know the muscle band is a garbage item. Please suggest a better item that's obtainable before defeating Lance.)
The 3 best Pokemon that learn a sleep move are probably Haunter, Jynx, and Lapras. I recommend not using Lapras because you already have a strong water Pokemon.
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source: same as the one that said Heracross is the best false swipe user
Haunter @ poison barb/choice specs
EVs: 100 SpA
- sludge bomb
- shadow ball
- hypnosis
- thunderbolt
Jynx @ twisted spoon/choice specs
Ability: forewarn
EVs: 100 SpA
- psychic
- ice beam
- lovely kiss
- focus blast
If you don't want to use your master ball, then I'm pretty sure the next best ball is the heavy ball. Lugia is over 204.7 kg, so the heavy ball raises its catch rate from 3 to 23.
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In addition to bringing a bunch of heavy balls, you might also want lemonades and revives in case Lugia damages your Pokemon. If Lugia has 1 HP and is asleep, and you throw heavy balls...
You have a 19.752% chance of capturing it per ball. Thus, you have at least a 50% chance of catching it within 4 balls and at least a 95% chance of catching it within 14 balls.
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That was a lot of sources.
Defeating the Pokemon League is a lot simpler than catching Lugia. It involves getting Pokemon with high levels, high attack stats, and strong moves and challenging the Pokemon League until you get enough experience to win. I think you should stop using Skarmory and Swinub in battles. Their levels are very low, and raising them is much more difficult than finishing the game without them. Feraligatr's physical attack is stronger than its special attack, so it would be a lot more useful if you taught it better physical moves.
Feraligatr @ mystic water
EVs: 100 Spe
- waterfall
- swords dance
- ice fang
- crunch
After you assemble your team of Feraligatr, Scyther/Pinsir, and Haunter/Jynx and teach them some good moves, just challenge the Pokemon League over and over until you win.
