First, imagine this situation. A car hits an empty garbage can and sends it flying. Would it make any difference if the car had an extra 100 pounds of weight in the trunk? No, the car is already so much more massive than the garbage can that any extra weight makes no difference, except to possibly slow down the car's acceleration, resulting in a lower speed impact.
There are three aspects of a swing that affect how far a hit ball will travel:
- The speed of the bat,
- The direction the bat is traveling at impact, and
- The location of the impact on the ball.
None of these will be helped by the backpack idea. The only effect will be on the speed of the bat. A swing consists of the combined motions of the leg muscles twisting the hips, the abdominal muscles twisting the torso, and the arm muscles swinging bat. You want your upper body rotating with speed and a weighted backpack will only slow you down or throw you off balance.
Without having seen your swing, I can only guess at what would help. But, one way that increases the distance the ball travels (using points 2 and 3, above) is to swing the bat so that the impact occurs while the bat is moving with a downward slope. This has two effects: it makes it less likely that you hit the very bottom of the ball, resulting in a pop up, and it puts backspin on the ball. This backspin gives the ball lift through the Magnus effect. It's the same thing that makes a curveball move sideways due to the horizontal spin. This lift will make line drives and fly balls carry further than if they had no spin for top spin.
Here's some further reading about batting coaches changing players' swings to turn them into long ball hitters: Why MLB hitters are suddenly obsessed with launch angles - Washington Post
Here, however, the batters are swinging up on the ball to get it in the air with power. This makes the location of the impact on the ball more important, as they only have the lower half of the ball to aim at. Too low an impact means a pop-up. Just below the center of the ball gives a good launch angle, speed, and backspin to carry the ball far.