The original Dash buttons were based on the Broadcom (now Cypress) WICED SDK, and as a result there have been several online writeups on how to develop custom firmware for them. Typically in such a system, the MAC address is merely a data item in flash at a location specified in a header file; hence it is something readily changed. (The newer ones use an Atmel embedded WiFi solution; likely something similar is possible, though I've not looked into the details, or it may be simpler to use an ESP8266 with one of Richtek's USB power switch chips to enable it to hold itself on just long enough after being bootstrapped by the button push to transmit).
So, if your theory about only the MAC address mattering is correct, then you could make an original Dash button or its functional equivalent transmit traffic from any desired address.
That may or may not be wise, but realistically, the greater concern is that the Smart Things system is hopefully looking for more than just "any" packet from a given MAC address, as such a system would be extremely weak. Also, Apple devices no longer use a consistent MAC address when merely scanning for AP's, in order to combat shopper fingerprinting systems which recorded these, making the seen scanning theory even more doubtful.
More likely or at least hopefully the Smart Things system is looking for actual background traffic from an App on the phone (or possibly some functionality of the phone operating system), with a cryptographic proof of identity or an account unique token protected by an encrypted channel. If it really only needs to see the MAC, that's weak enough that you may want to rethink what you allow to be triggered by an at-home detection.
In terms of adding "a fake mobile phone" you might be able to accomplish something if you can get their app to run on an Android emulator or a more engineering sort of Android device platform for which you've configured the MAC address. Or in the more likely case where they need more than the MAC address, such a device that you put into active mode by pushing a button.