That quarter ounce
Won't make a piddily difference. A good player can shoot with any cue, heavy or not, even though it may not have the feel and balance and weight that they prefer. If you shoot soft all the time, it tells me you are not real confident in your shooting. You would probably do yourself better by learning better cue ball control, or at least shoot a little more towards center ball if you are over running shape.
I have lifted weights over 40 years and know the effects it can have on your game. First, DO NOT lift on days you are playing Pool. Arrange your schedule to lift on 'off' days. On inbetween days of lifting (Pool days), do stretching exercises to loosen back up your shoulder and back muscles and arm muscles. Stretch at least 2 hours before going to play pool. and don't overstretch, your relaxing those muscles from the day before workout, not trying to tighten them back up.
The most trouble I had from lifting was I would workout and my muscles would contract, making me feel strong, and I would 'overpower' the break, jumping the table frequently. Plus a few certain type of shots would be a little off for shape, which in my case, could lose the game or match for me.
I DO NOT work out with heavy weights, although I worked with heavier weights when youinger. I have geared it back as I have aged. I only work with 50 lbs now, as I am just trying mostly to maintain strength and keep toned up. Overworking your body with heavy weights can cause you all kinds of problems with your Pool game.