You can't hurt anything doing this so I would give it a try. Put a resistor on one of the terminals of the Elk relay, then hook your one lead from the garage door up to it and the second lead directly to the other terminal of the Elk relay. So now, when the Elk closes the relay, it will close the circuit with current flowing through the resistor. The resistor will put enough "back pressure" on the current so that some current will still flow through your smart button keeping it from rebooting.
The question is, how much resistance? If the resistance is too much, the garage door won't recognize the closing of the circuit and won't activate the door. Too little resistance and the smart button looses too much juice and resets.
I assume you have resistors sitting around that came with the Elk for use as end of line resistors. It won't harm anything to try one of those. If closing the circuit without a resistor didn't damage anything, closing the circuit with a resistor certainly won't.
Thanks for the tip Lou Apo and everyone else. I have decided to give BSR's suggestion of hacking a wireless remote a try and I was up and running within 15 minutes. I will definitely revisit the resistor thing in the spring time when it is not -30 Celsius (-22 Fahrenheit) out as I would prefer a 100% hard wired solution, but this will get me going for now.