Heat 2 teaspoons oil in a skillet over medium heat and cook fennel and onion for 8 minutes until beginning to soften. Transfer to a 6-quart slow cooker with broth.
Mix remaining oil with garlic, rosemary, thyme, salt and pepper and rub over pork loin. Place over the vegetables, cover and cook on low for about 3 to 4 hours, beginning temperature checks at 2 1/2 hours.
When the roast reaches about 135°F, spoon drained beans into the hot liquid around it and cover again. Cook until the pork reaches 145°F in its thickest center and the beans are steaming hot, generally another 15 to 30 minutes.
Lift pork to a board and rest for 10 minutes. If fennel or beans need more time, cover the cooker on high during this rest until vegetables are tender and beans are hot throughout.
Stir lemon juice through the bean mixture. Slice pork across the grain and serve with fennel, beans and a little cooking broth.