Smart switches are not equal partners like old style 3-way switches. They are "master" and "remote".
This model of smart switch is a "master" and requires certain models of "remote" which require "neutral" (white wire) to be present in both switch boxes. That is not always so, as the electrical code only recently started requiring it.
So I'd first check the other box - if there's no neutral in it, you cannot wire a remote (though wireless options may exist). Useless old work can be dangerous, so tidy up by removing the 3-way switch, wire-nut the common to one of the messengers (trial and error will reveal which), cap the other one off, and put a blank wall plate over the hole.
If a white wire is already wired to a terminal on the old 3-way switch, bad news - that is NOT a neutral, but one of the messengers (the two inter-switch wires). Before the code change, they often used common 14/2 Romex for messenger, and one of the wires is white. They're supposed to paint or tape it.
If there is a neutral in both boxes, you can wire a remote. First, figure out which box is upstream (closest to the service panel aka breaker box). It will contain a wire that is always hot - regardless of the position of any switch.
Then review the diagram (which you already linked) showing you the wiring scheme for whichever remote you bought. You will need to re-use both of the old messenger wires.
I have the same smart switches and it was a breeze to install. Have you downloaded the Kasa app on your phone? Once you open the app and select new install, select the diagram that represents the currently installed switch. After that it gives step by step instructions for proper installation.
Best Answer
What you have is a plain switch, a dumb 12VAC transformer for the pool lights (finally, someone who takes pool safety seriously, yay!) and several "smart" LED lights with a trick. They can change colors if power is interrupted for a moment. And they can all do this in sync, which is usually the hard part.
You are now implementing a "connected home" platform, and you'd like to control these LEDs. You're looking for some sort of glue/middleware device that will let Wink address the LEDs and change their colors. Nope.
Not supported
You can put a smart-switch on their supply, sure. You can have the Wink switch them on and off, sure. Wink might even be able to turn them on and off quickly, as needed to switch their colors. But Wink won't understand what it's doing or why it's doing it.
There won't be a way to tell Wink "make the pool orange" and have it tip-tap out the command to make those lights orange. There may, but it would involve some special software coding; the question is whether Wink gives you the platform/language to do that. Such a command series would not remember where the lamps are now, and certainly can't see them, so it would have to start the sequence from a cold reset-to-white, so it would take 10-15 seconds to reach any particular color. On the other hand, Christmas light controllers do that kind of thing for breakfast, so maybe.
This is a hard problem, and it is unsupported: there's nobody to call for support to make this work, as you are discovering.
Though, given the cost of these things, it might be worth contacting the factory. You can bet this has come up before. They might even have a solution, or be motivated to develop one. Their customer base are exactly the sort to do smart-home tech.
Or start over
One could go with LED lights designed to work with Wink -- or alternately, a controller designed to work with Wink. I don't think you care if they're individually addressible, so that's a plus. The big minus would be if this forces you to 120V lights - because you are quite correct to keep 120V away from your pool. Electricity is much more lethal in water, because you can't pull your hand away, and it only needs to stun you to drown you.
The good news is if you can get a Lutron dimmer that plays with Wink, maybe you can find an RGB controller that works with Wink too. I have started to see them from name-brand companies, so hopefully. Those controllers are designed for LED strips, input 12 volts DC, and output 3 channels (4 wires, R G B and common) which go direct to simple LEDs. That would require you to run 4-wire cable down to each of your pool light locations (I sure hope they're in conduit). And thermostat cable would suffice since you're at 12V.