Its best to focus on one goal at a time rather than stretch goals. If you were learning to do ballet, you wouldn't also try hip hop and painting in the same day right?
Fat loss benefits from weightlifting, but it plays a better role in cutting fat and maintaining muscle. If you're trying to soley lose weight, switching between high, moderate, and low intensity cardio exercises throughout the week as well as switching up types of exercises along with diet will burn the most calories. You can also do HIIT.
If you're trying to maintain hard built muscle but lose fat, then I generally do one strength training full body day, and two weight days where I do full body strength training but metabolic circuits or circuit training/endurance workouts, then I add a moderate cardio day, 1 high cardio day, and walk twice a day for 60 minutes(low cardio).
Otherwise for hypertrophy, strength, power, etc focus on the weights first and do cardio after your workout as an afterthought.
If you're bored with cardio, you can always do body circuits metabolic training which is essentially HIIT, or HIIT, sprinting, etc