First one quick question. Do you get any smoke out of the stack as the turbine spools down to a stop?
The -112, Conquest I and maybe the Caravan? Not sure. Anyway on the compressor wash. First I am not a PT6 expert by any means. I do read a lot and the mechanic that did all of the maintenance on the plane I just retired from flying is an expert. I was present for all maintenance and did not shy away from asking questions. In fact we became good friends and still talk frequently.
If you are flying in salt areas, coastal or you are flying at low altitudes (Caravans, ag work for example) you should follow P & W washing instructions. The salt can combine with the sulpher in fuel and lead to sulfidation. Also the low turbine temps can cause more carbon build up. I will skip this since it does not apply to you.
If you are doing a "compressor performance recovery wash" then a rinse of drinking water quality water must follow this. Both are to be done a 10-25 %Ng
Here is where there is a difference of opinion. The CT and PT blades slide into their respective hubs. Some say the soap used in the power recovery wash gets into the spaces around the blade to hub area and is difficult to wash out causing corrosion in that area of the blades. This would be exasperated if the cleaning is done and the engine sets for hours or perhaps days before it is run. My mechanic said that the PT 6 run at the proper temps should not have any build up even at several hundred hours. My engines were the -61's which are somewhat bigger than yours but should be the same principle. We never did compressor washes. These particular engines were 24 years old with just under 4000 hours at OH and had no sign of sulfidation or corrosion.
Again, not giving advice just throwing out ideas. If this happens again I would want a borescope done by somebody that has the correct attachments to do the Pratt engines and see if you are getting carbon build up. If you are, something is wrong. I promise you these engines do not resolve their problems on their own.
I am not familiar with the -112 engine but, I do have a question. I believe their thermodynamic rating is just 500 HP. Are they flat rated to a lower HP? If not it would seem they would temp out way before 16,000 feet. Not related to the OP, just curious.
Edit: I do not think there is anyway to check the thermo couple ring itself. At hot section they may have been replaced. There is a trim stick that the overhauler uses to get the the thermocouple to give the correct temperature. This is normally checked at overhaul on the test stand. It is all kind of magic to me. All the mechanic can do is inject a millivolt signal into the leads at the engine and see it they read correctly. Any corrosion at any junction can throw this off. As far as I know you just have to hope the thermocouples along with the trim stick is correct. IMO if you have accurate fuel flow this is the best way to match engines. I know we are all told to match torque. The torque is actually an oil pressure sensor and can be a little flaky. I found it hard to get the torque gauges to read linear full scale, YMMV. If you happen to have digital fuel flow that you know is accurate I think the matched FF is better. This is all just opinion and may not be worth anymore than you paid for it.