There are two things going on with your weaker side breathing, 1. recovery arm/elbow is lifting above your back causing imbalance (sink) and 2. the low side edge is not as balanced and stable, largely due to 1. noted previously.
So on weaker breathing side focus on 1. swing recovery arm wide, away from body line and lead with elbow -high elbow will happen naturally, 2. practice/refine low side skate edge (edge opposite breathing shoulder) or what I characterized earlier as "broccoli side".
The single arm drill will not help much and only cause you more frustration integrating breathing into your stroke. Instead, practice breathing in skate drill. Once stable in skate - rotate head until you find air off shoulder, then return to goggs down. Don't bend the spine and/or lift head to find air from skate, but rather learn to stabilize your skate position until you find air off your shoulder.