I need a shoe for hiking several miles in the heat that has sticky rubber for wet rock and comes in size 14. I've found that approach shoes are too hot and heavy, and the rubber on trail running shoes isn't sticky enough. Worse, brands like Sportiva and Salewa don't carry anything that fits.
That's a tough question, Scott. Once you get into size-14 territory your options are slim. Specialty shoe makers aren't wild about making lasts that only a few dozen people will buy.
I'm surprised that you've found approach shoes to be hot and heavy. A shoe like the Scarpa Epic ($95, comes in 14) is light and feels almost like a hybrid between an approach shoe and trail runner. It has a rubber toe rand for protection, mesh uppers for breathability, and a lugged, sticky outsole.
Montrail's Mountain Masochist ($95, goes to size 15) is more of a pure trail runner, but it has a nice, grippy sole. Like the Epic, it doesn't come with a waterproof lining, which makes it a lot more breathable.
If neither work, you could even look into a fishing shoe like the Simms RiverTek BOA Boot ($170). It's a light hiker that is designed to be in and around water and slippery rocks. They're high tops, however, so even though the uppers are made of breathable synthetic, they're still apt to feel a bit warm. And they aren't really meant for hiking.
You could even try a slip-on like Muck's Cikana Shoe ($80). They're light and comfortable, and designed for wet and slippery conditions. They're made from a neoprene-like material, which isn't breathable, but they are low-cut. And you always can douse your feet in a stream!