I suspect that your rim tape isn't moving by itself. Like most inanimate things it prefers to stay put until moved.
Most likely you're moving the tape when you push the tire across while mounting the second side.
Maybe, you can completely remove the tape, use acetone (not nail polish remover) to clean the rim's belly to clean, dry bare metal. Then apply a new rim strip, tensioning it as you apply it, then burnish it down so it sticks well and stays put. Or you can try your luck with one of those stretchy rim bands, or filament tape. Both have the advantage of a more slippery surface so will be less likely to get pushed over when you mount a tire.
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