I'm not sure a split path would work, especially on the busy sections from, say, the fun park to the baths at least. I think cyclists (and I'm one, most days) just have to learn to adapt to the conditions. Slow right down, watch for kids, give way, and, ultimately, get off and walk when it's busy. The (slightly) less vulnerable cyclist should always give way to the more vulnerable peds.
Since the prom is now a core path my understanding is your legally obliged to behave 'responsibly', as defined by the terms of Scottish Outdoor Access Code.