Preheat the oven to 400 degrees F and line a rimmed baking sheet with aluminum foil.
Heat a large high-sided cast-iron or oven-safe skillet over medium-high heat and add the vegetable oil. Pat the pork tenderloin dry and season with the smoked paprika, 3/4 teaspoon salt and 1/2 teaspoon pepper. When the oil is shimmering, add the pork and sear on all sides, about 5 minutes. Transfer the pork to the prepared baking sheet and roast until it registers 145 to 150 degrees F on an instant-read thermometer, 20 to 25 minutes. Let rest for 5 minutes.
Meanwhile, set the same skillet over medium heat, add a drizzle of oil if needed and add the peppers and onions. Add a splash of water to deglaze and use a wooden spoon to scrape up any brown bits from the bottom of the skillet. Cook, stirring occasionally, until the vegetables begin to soften, about 5 minutes.
Add the garlic and cook until fragrant and slightly softened, about 3 minutes. Remove and discard the tough, fibrous ribs from the kale leaves, then roughly chop the kale leaves. Add the kale and use a wooden spoon to carefully fold the hot vegetables over the kale to help it begin to wilt. Continue to cook until the kale is bright green and wilted, 3 to 4 minutes. Stir in the white beans and a pinch of salt and pepper. Cook until heated through, about 2 minutes more.
Slice the rested pork on the diagonal and serve with the white bean and kale mixture.