Peel pineapple, then cut crosswise into 1/3-inch-thick slices. Cut slices into small triangles. In a large shallow bowl, place pineapple, rum and 2 tablespoons sugar; toss together to combine. Let stand at room temperature, tossing occasionally, 1 hour.

Drain pineapple and pat dry well. In a large shallow bowl, whisk together eggs and pinch salt. In a second large shallow bowl, stir together breadcrumbs and remaining 2 tablespoons sugar. Dip 1 pineapple triangle into egg, letting excess drip off, then roll in bread crumbs to generously coat, pressing gently to help crumbs adhere. Transfer to a large plate. Repeat with remaining pineapple.

Line a large plate with paper towels. Heat about 3 inches oil to 350° in a 4- to 5-quart heavy pot over medium-high heat. Working with about 6 at a time, drop pineapple pieces into oil and fry, turning once, until golden, about 3 minutes.

Using slotted spoon, transfer fritters to paper towels to drain; sprinkle with light pinch salt. Serve warm, with honey for dipping.