OK I see the issue now
Two suggestions:
(1) There are a number of ideas in this thread on how to turn voice control on/off on the Mac. It can be done via an Applescript; you can save the Applescript as an App.
Could you write one Applescript which turns on your Dragon dictation and turns off Mac dictation. Then another one which turns on Mac dictation and turns off Dragon dictation?
(2) Do you have an iPhone or iPad with accessibility features turned on, i.e. the Grid to control the iPhone/iPad by voice? If so then you could use the iOS version of Keyboard Maestro. With it you could use voice control on the iPhone or iPad to execute any Macro currently on Keyboard Maestro on your Mac. I suspect this is the most fail-proof solution. One of the macros you execute by voice on iOs could even be an Applescript to turn Mac voice recognition on or off.