No, insulin doesn't cause weight gain. I want to shout this very loud because it's at least the second time I've said it today! Put that **** out of your head. Insulin saves lives and preserves limbs, kidneys and eyesight. Insulin does not make you fat. If you eat a healthy diet and take the amount of insulin you need to control your blood sugar, then that's good. Please don't be afraid of insulin.
You'll still need to bolus for protein if you're eating LCHF. It's not an easy answer control-wise. Have you tried using something like MyFitnessPal to count calories, carbs, etc? Calories can matter too if you're struggling to lose weight. Keeping records will show you if there's any obvious answer to your difficulties in losing weight. LCHF doesn't suit everyone. Some people end up having to take large/larger amounts of insulin because the diet itself can pitentially mess with your insulin sensitivity.
No, a pump wouldn't make you put weight on. It would just allow you to tailor your insulin more closely to your needs
I sympathise with your weight worries. Sorry if I've said all this before, but have you had your thyroid checked, could there be a hormonal issue, have you asked about Metformin? I've probably said at least one of those before, but they're all worth repeating anyway.