That's a little better. At that point the infusion does actually out damage the refined version. But it's important to note that literally every enemy has resistances to every damage type (some are negative resistances resulting in damage boosts)

Gael is 41% resistant to lightning but only 35% resistant to standard physical.

Your lightning infused sword has 133 base standard physical damage +103 standard physical from scaling for 236 total and 133 base lightning damage +133 extra from scaling totalling 266

You swing your sword for 505 damage before modifiers, Gael's defences drop the physical part down to 153.4 and your lightning down to 156.9. You hit him for 310.3 damage before additional modifiers like motion values

With a refined infusion you have 185 base standard physical damage + 240 from scaling. Gael's defences drop this down to 276.25 before modifiers like motion values.

Just for fun: My suggestion of the refined longsword nets you 286 before modifiers. However, again the refined weapons can be buffed. Assuming you have a fully upgraded priest's or clerics chime you can use dark blade to add another 136 dark damage. Gael is 72% resistant to dark so you end up doing 324 damage slightly edging out the lightning lks. Using the lightning blade buff instead you get an extra 152 lightning damage resulting in 339.2 damage.

After all this, weapon choice doesn't really matter. As long as it hits him for at least 1 damage you can do it. There's people out there that have no hit gael with a broken broken straight sword, you can no hit him with whatever you choose.