Table 310.16 only goes to 176F ambient, so I don't think that you could use 310.16 as the basis for operating THHN in a 180F environment.
However it is pretty easy to calculate that if the table had an 81-85C row, then the correction factor for THHN would be 0.29 .
This tells me that in a 180F environment, you could have on the order of 8 or 9 A flowing through 12ga THHN conductors without overheating them. But I think that you would need appropriate 'engineering supervision' to make the installation code compliant. I would also imagine some difficulties with design of the over-current protection.
Additionally, while THHN is rated for 90C, I would not be surprised if the conductors themselves can easily tolerate higher temperature. I regularly use wire that is rated as THHN and THWN and MTW; with a 90C rating as THHN but a 105C rating as MTW.
The big thing that would make me uncomfortable is Bryan's question: how well can you trust the 180F number? If the real value is 190F then the ampacity of the conductors is reduced still further, and if the real value is 200F, then the insulation would be overheated with no current flow at all. The other thing that makes me uncomfortable is that you are getting so close to the bleeding edge of the insulation rating that small errors in calculation or conditions might use up all of your safety margin.
-Jon