I am not a pro accordion tuner, and I am not aware of any devices made specifically for tuning accordions.
I am, however, a guitar luthier and I know a thing or two about tuners in general, albeit none of that knowledge is particularly helpful when it comes to tuning squeezeboxes.
The only "special device" that I can think of is a strobe tuner. It was a real breakthrough in precise tuning technology decades ago, and these days you can buy a digital strobe tuner that fits in your pocket but mimics the look of the analogue one - very successfully. These tuners did not really help me when it comes to tuning reeds.
There's no need for a high end microphone either. The tuner is not interested in the whole spectrum - it's just listening to the main frequency. Unfortunately, I'm too old and forgetful to remember how exactly that is done when it comes to analysing the frequency peaks - probably, just finding the highest peak.
The stuff you find in mid-range mobile phones is good enough these days.
There are many correct ways to do the job, you should listen to what they tell you on the course, try several and see which one works for you.
I have followed Debra's advice to use a mobile app called DaTuner which seems to be working out great.
To me, the real challenge with tuning professionally (not something that I ever intend to do) is nothing to do with the equipment, but knowing when to stop. That is, I can work on a box for a whole week and it will come out sounding incredible, but I'll struggle to find a customer who will be happy to pay for a full week of my time. How close to perfection do you take it, before the next installment of accuracy starts costing more than a regular Joe is prepared to spend on a musical instrument? You want a steady flow of customers and you want them to be happy, so it's a balancing act of how deep down the rabbit hole you go.
With lutherie, there is always another guy down the road, who will do a third of the job for half the money. He will make your (high quality) services look overpriced, and a lot of customers will go to him, waste their money, end up being unhappy with the job, and result in you losing out on extra income. Your main asset is your reputation. It's a tough balancing act, and a very tough occupation if you intend to have it as your main & only job.