Try clearing the memory and follow the steps below. If it is a rolling code gate then you will probably never get it to work, not sure why it is working in the Lexus (they are fancier so that could be it). I may have read your post wrong but it sounds like $35 will fix this and get someone to press the learn button on the logic board of the gate for you. If this is the case then I personally would just pay it and save myself the aggravation.
To clear the memory of all three buttons in the HomeLink transmitter, hold down the two outside buttons (1 & 3, 2 being the one directly under the house logo/light) for approx. 20 seconds until the red light begins to flash rapidly. Release both buttons.
Phase 1
Be sure people and property, including yourself and your vehicle are NOT in the travel path of the garage door and that nothing will be struck when the door attempts to close or open !
Turn your vehicles ignition to the "Run" (or "accessory") position.
Hold the transmitter 1 - 3 inches away from the HomeLink buttons in you vehicle (do not block your view of the HomeLink indicator light)
Simultaneously press the transmitter button and the HomeLink button you wish to program (you may need two hands). Hold both buttons pressed until the indicator light changes from a slow flash to a rapid flash.
Rapid flashing indicates successful programming - Release both buttons (HomeLink and transmitter)
Press and hold the newly programmed HomeLink button,
If the indicator light is solid (not flashing) your garage door should now activate and you are complete. There is no need for Phase 2
If the indicator light flashes for 2 seconds before turning solid (not flashing) your garage door opener is rolling code technology and you must continue Phase 2 of these instructions.
Phase 2 - Only perform the following steps after successfully completing Phase 1.
A second person will make the following steps quicker and easier. Be sure people and property, including yourself and your vehicle are NOT in the travel path of the garage door and that nothing will be struck when the door attempts to close or open !
Locate the "Learn" (or "Smart") Button on the motor-head unit of your garage door opener. (This button is normally located on the back side, near the hanging wire antenna. Consult your manual for exact location.)
Fully press and quickly release the "Learn" Button (holding this button pressed for too long will erase all existing transmitters and keypads and they will need to be reprogrammed into the unit)
Within 30 second of pressing the "Learn" Button, return to the vehicle. Press the desired HomeLink button for 2 second, then release. Repeat this one step up to 3 times.
Your Garage door opener should now activate with a single press of the HomeLink button