How close are those LED lights to the door opener itself? Radiated RFI (Radio Frequency interference) from the LED lights may be coupling into the receiver of the door opener and masking the desired signal transmitted by the button opener. In this case the interference is "desensitizing" the receiver, in other words reducing its sensitivity level. If this is happening, then it's likely that moving the button very close to the door opener will overcome the level of interference and allow it to operate. It might be worth doing this as a test.
There could also be conducted interference from the LED lights through the 120V wiring. Are the lights on the same branch circuit as the opener? This could make the situation worse. Another test would be to insert a Tripp-Lite or other power strip with RFI filtering before the door opener to see if this helps. If it helps then conducted interference is an issue and a line filter could be a solution.
Conducted interference from the LED lights could also be radiated by the branch circuit wiring. It's really the job of the light fixture manufacturers to provide adequate line filtering. Technically line filters could be inserted in front of each LED light but I'm not sure if there's anything available that is suitable in this case and compliant.