Fill a deep pot with 3 inches of oil and preheat to 350˚F.
Place the dredging mixture into a shallow dish and whisk together.
Pour all batter ingredients, except the seltzer water, into a large mixing bowl and whisk together.
Place the squid, shrimp, fennel, potato, and lemon into the dredging mixture and toss together until everything is fully coated in a thin layer, shaking off any excess (you may want to do this I a few batches).
Pour the seltzer water into the batter and whisk together. Dip each dredged item into the batter and shake off any excess.
Carefully drop coated vegetables into the oil, first a few at a time, making sure not to overcrowd the pot.
Fry each piece for 3 to 4 minutes on each side or until very lightly golden and crisp. Drain onto a paper towel and lightly season with salt. Repeat with the remaining vegetables followed by the seafood, and finally the lemon slices and basil leaves (the basil leaves will only require a couple seconds of frying).
Once all the items are fried, transfer to a serving platter and serve with garlic-lemon aioli and cocktail sauce.