Before reading all of this, try switching out the LED's for incandescent and test it.
I had a similar problem in my Gen 3 this fall. Cruise control disengaged when I put on a turn signal. This may be like yours, but it may not. My apologies, I would make my response more concise if I had the time.
Quick summary of my solution: bypass the electronic towing package if you have one and check for burnt out or dim lights in the rear of the vehicle. Replace LED tail lights with regular incandescent or high wattage LED's that better match the impedance of incandescent lights.
Here was my problem and solution:
Once upon a time, I had a working electronic braking system. It's an add-on towing package that engages brakes, turn signals, and brake lights on a trailer when I tow with my 4Runner. Then, I towed a trailer that was mis-wired, and my brake controller caught fire and failed. Several years later, a short developed between the turn signals and the brake wiring in the towing package. When I changed lanes while the cruise control was engaged, the cruise control would disengage when I turned on my turn signals or caution lights. This is where it gets ridiculously complicated.
The electrical short passed a low voltage (~6 Volts) from my turn signals to the towing package's brake system. The electronic brake controller would activate and trip the cruise control's brake sensor, so the cruise control would be disengaged. The brake controller actuated with the blinking of the turn signals. In addition, the electrical short caused the 4Runner's red LED brake lights (not the running lights/circuitry) to flash in sync with the turn signals. The right turn signal caused both of the LED brake lights to flash. The left turn signal only caused the right brake light to flash, unless the right turn signal light was unplugged. If the right turn signal was unplugged, then both brake lights would flash in-tandem with my left turn signal. There was some weird electrical black magic happening then.
After three weeks of trouble shooting, I deduced the short was between my right turn signal and the braking system. By powering the right turn signal with an independent power supply, I was able to replicate the failure. Unfortunately, that power supply had enough current that a loud "POP" was heard when I turned it on. I'll get back to that later.
After the "pop," the ABS failed and was no longer functional. Turns out that the ABS system turned off to warn me that my cruise control had somehow failed. However, after the brake controller was removed, the cruise control worked flawlessly and did not disengage when I turned on either of my turn signals. The cruise control properly disengaged when I hit the gas pedal or the brake pedal. The cruise control was now normal. The ABS was dead.
Ironically, the ABS system turned on again if the cruise control was unplugged. Because of my independent power supply stunt, I somehow damaged the cruise control with the electrical short that traveled from the right rear turn signal through the rear towing package to the dashboard electronic brake controller into the cruise control's brake sensor. Technically, the previous sentence isn't a run on. Anyway, back on topic. To figure that last bit out, it cost me $600+ parts and labor for a mechanic to fix what I screwed up by fixing my turn signal-disengaged-cruise control problem. If my LED 7440/7443 tail lights were not dimming/dying of old age, my cruise control probably would not have failed during my testing.
Moral of the story? Replace all tail lights with C5 Cree (no SMD's) or incandescent bulbs and bypass the electronic brake/towing package if the cruise control disengages with the activation of the turn signals.
Watch the little red LED flash on the dashboard tow-package brake controller. It normally flashes when the driver applies the brakes. It flashed in sync with the turn signals, helping to reveal the electrical short.
