I've heard that weight on the wheel matters more than weight on the frame (or shoulders, or jeans).
Everyone else who is giving you advice in this thread has heard the same thing, but they're choosing to ignore it.
It does mater more, but not so much for climbing. It matters most for quick acceleration because there is forward acceleration of the total mass of the bike and rider, and also rotational acceleration to spin the wheels up to speed. For climbing, total mass is extremely important, but the significance of rotating mass VS total mass is diminished when you add the force required to lift the bike/rider into the equation, the rotational inertia of the wheels is a drop in the bucket.