Here is any easy solution.
Purchase an auxillary reciever unit for your garage door opener, and wire it in parallel with the button I'm sure you have beside your back door. Make sure to get a cheap one without rolling codes, etc. Here is are some examples, about $45.00
or this one:
Then you program your Homelink unit to match you new receiver's code. Bonus is that your old remote will still work too (pass it off to your S.O). Downside is that you lose the "rolling code" feature when you use the Homelink unit.