One concern with a Thermos, would be the thermal shock of adding liquid nitrogen might crack the glass liner. If you use steel thermos, I don't know if they would also be damaged (steel expands and contracts quite a bit with temperature changes). The containers we use are steel outers with a thick silvered glass liner. Judging by the radius of the glass as it bends at the top, I would say the glass is 3-5mm thick. When I last broke a regular thermos, the glass was very thin indeed - I wouldn't trust it to hold.

Remember LN2 is very cold, so the change of temp from room temp to liquid is about 200 C, this is a lot more than adding boiling water to a Thermos. You might want to try putting your open thermos in the freezer for a couple of hours, then adding boiling water and see if it holds up. If you aren't confident of doing this, then don't try it with LN2.

Also, when carrying LN2, you MUST vent the container - never seal it shut. The change from liquid to vapour is massively expansive, your Thermos will simply explode in short order, or boil over you as you open it. Don't leave it in small, poorly ventilated spaces, the nitrogen will displace the air and you will asphyxiate - nitrogen is 80% of normal air, so it doesn't smell, nor is it visible.

Overall, I am not sure a Thermos is a good solution, too many things might go wrong, thin glass is brittle enough without being held at such a cold temp.

I have not discussed the advisability of using LN2 for DIY dermatology - but you should exercise extreme caution when handling it, and I would recommend having adequate medical and technical supervision. When used properly however, it does yield extremely smooth ice-cream!