Wearing any HRM (heart rate monitor) while weight lifting is a pointless effort. Weight lifting does not = cardio, which is what all these HRM / trackers are calculating.
Steady state cardio is what all of the algorithms are focused on calculating. Also, if you have one of the older Fitbit trackers that does not incorporate a true HRM, it is just a dumb guess and does not incorporate any actual output / exertion on your part.
My advice is to buy a nice Bluetooth HRM Chest Strap like the Polar H7.
Amazon.com : Polar H7 HR Sensor BLE BLK M-XXL USA : Heart Rate Monitors : Sports Outdoors
Download Endomondo app and pair the BT HRM with your smartphone. It is not only the cheapest option, but performs better than most of the outdated smart watch fitness trackers and it's quiter versatile. Endomondo has tracking for just about any activity you can think of, has a great web interface, tracks lots of things like your personal bests, social connection with friends, group challenges, local routes for running, biking, etc.