From the wires, looks like the left side is input, and the right side is output. A picture of the back of the circuit board might be more helpful here.
It’s likely the LED uses serial control for color and brightness. So, the 3 wires could be positive, negative, and data. If that’s the case, maybe replace the whole setup with a control pod like WLED if it supports the protocol. Here’s one option that’s compatible with both 5V and 12V systems, or try this USB model with a button.
You could simulate the button functions with logic switches that replicate the button actions. DM if you need a schematic.
Depending on your skills with electronics, you could attach an Arduino-like ESP8266 as a relay with the GPIO simulating a button press. They’re just momentary switches, so maybe solder some leads to the board, but keep in mind there might be more electronics on the back. Here’s a project like that in this video.
I made a siren ‘smart’ with a similar button setup using ESPHome and a Wemos D1 with a relay.
Can you post a pic of the PCB back? There should be an MCU back there. Would help a lot.
Ever thought of wiring a microcontroller into the buttons? I did that with a cat food dispenser. If the buttons connect to a small controller, you could add a small ‘spy board’ to simulate button presses.
Could get bulky, but you might try a 4-channel Zigbee relay from AliExpress to keep the buttons as-is and control them.
Using optocouplers and an ESP could work for your setup.
Easiest fix? Just buy a lamp that uses regular bulbs and add a smart bulb.
To get it working the way you want, try moving the smart switch to control the button functions, not the AC power.
Low-voltage DC relays and a microprocessor would be needed. Do you use the color and dimming options? If not, you could remove the switch and power the LED directly with a different power supply if you know the correct specs.
Try sending it to college… I hear that helps.