Your intuition to question this information is right on. Your trainer is mistaken--calorie consumption during weight bearing exercise is directly related to total body weight.
Think about it, who would burn more calories: a hiker carrying a small day pack or a hiker carrying a 40-pound backpack? Certainly the backpacker would have a much higher caloric consumption. Body fat weight has the exact same effect on caloric consumption.
Body weight has varied influence on non-weight-bearing activities like stationary cycling. There the caloric consumption depends on the workload not on the body weight. Remember, calories reported by the exercise machines are guesses at best.