A Patong-Kathu shortcut does exist, but you would definitely need a high-clearance 4WD vehicle or a small and light motorbike that your can maintain control of while sliding around on the steep and muddy hills that you must traverse along the way. Once it dries out, it will be a lot easier. I talked to a woman on the Kathu side that walked the road and she verified that it was possible but said that I should not try it unless I had better shoes. I was wearing flip-flops. So I rode my bike around the one hour detour through Kata/Karon to Patong and up to the top of the mountain. There I met 2 separate persons that had ridden small, automatic motor scooters up from Kathu and they asked me if they could get to Patong if they followed the road that I had just come up on. I happily told them yes and they each took off down the road smiling.
Going from Patong to Kathu, here is a picture of the first slippery and steep hill:
The steepness never seems to actually show in the picture. But be aware there is a surprise ditch in the road just as you come to the top of the hill, just big enough to upset the progress of the bikes with the smaller wheels.
Shortly after that, you will come to a fork in the road. Google says to go left at that fork. But knowing I already had one steep and slippery ass hill that I had to go back down, I turned around at that point. If I would have had actual shoes on, I would have traveled further. Picture of the fork:
Lucky my bike has semi-dirt tires on it, so I made it safely back down the hill. I have always enjoyed climbing hills on my bikes, but always hated going back down the particularly hairy ones.
The route is plainly marked on Google maps if you look for it. It runs between the Wat Doi Thep Nimit Monastery above Patong and the เคียงเขา Forest & Cafe at the bottom of the hill on the Kathu side.
Me, I will wait for things to dry out a bit and then I will put on my boots and go for it. You guys go ahead and report back.