I see David. If "hands-free" is not a pre-requisite, then what you have in mind is clearly viable.
In fact, I would go so far as to say "don't worry too much about it". By being a broadcast engineer, in all likelihood you have way more knowledge on the ins and outs of sound equipment than most frequenters of this forum, myself included.
From my own practical experience, moderate noise is no real problem these days. My ER colleagues use voice recognition to produce their notes, oftentimes in the middle of a busy ER, on a laptop. In such environments, I am now tempted to think that the limiting factor due to ambient noise/voices would be own brain's inability to focus, rather than Dragon itself.
This is not to say your recognition success doesn't decrease in a noisy environment. It does. Just a bit less than I would have thought myself.
And as "Bad Dog" alluded to above, monotonous background noise such as loud fans in the OR, or even a vacuum cleaner, are easy to "erase" with today's DSP. Zoom and Teams can do it with no seat. See also the thread on Nvidia "RTX Voice" noise cancelling technology, as but another example.
In my immediate environment, folks just use the run-of-the-mill PowerMic III or (IMO) better constructed SMP (which you are already looking into).
Moreover, to any extent that you could modify your environment, you could achieve significant mitigation. I know that once they erected those huge plexiglass shields (which was just a 180 degree flat "wall" and not even a 270 degree partial "enclosure"), our ER colleagues were already getting a better experience.
Finally, many of those cardioid-this-or-that patterned microphones take advantage of a "proximity effect", which others would know/explain better than I could. Basically the voices from a distance that enter your ears relatively clearly don't really get picked up by such mics that much.
As far as I know, some of those so-called "noise-cancelling" microphones don't actively "cancel" any noise. They are noise-isolating or passive noise-reducing. To get active noise cancellation based on directionality and timing of sound arrival, one would need at least two mics, preferably an "array". The details we will leave to real engineers.
Some of those more expensive, "call-centre grade" headset mics have dual mics with some active noise-cancellation. I am not even sure whether the SpeechWare FlexyMike DEC has any active cancellation. My guess is that it does not. But does it matter in your environment? Hard to know without testing.
But bottom line, in a moderately noisy environment, non-hands-free use of voice recognition with today's Dragon is eminently viable. It is done throughout your neighborhood hospitals every single day. 24 hours a day.
If you can modify your environment, such as by putting up some plexiglass barriers on 2 or 3 sides (make them as tall as they will allow, beat the Covid horse to your "advantage" ... after all, it beat us for 2 years), I'd say you are ready to tackle your goal.
The other piece of advice. Make your noise-dealing/minimizing endeavour SEPARATE from getting Dragon work for you. Tame the Dragon with the help of this forum in a quiet place at home first, possibly using just an "average" microphone. That way you'd know what to benchmark against.