PCB coating for a high temperature environment

This is well into the territory of "hire a specialist design firm". It's going to be expensive, so you might as well do it properly and hire people who know where all the pitfalls are and where to get all the parts.

TI have some high temperature microcontrollers, and a design guide linked from that page. The SM320F28335GBS is good up to 210 degrees (which is cutting it a bit close). While you can buy from Digikey in single quantities, they cost $300.

The other approach you could take if the operating time is limited to a few hours in that environment and the box can be sealed with no wires in or out, is to insulate it very heavily. Add some thermal mass, preferably with a phase change. Do not exceed the stated cooking time.

You'll end up with a something like a meter cube of fiberglass or foam insulation containing a sealed shoebox of methanol with the electronics floating in it, which will stay below 70C until it boils dry, hopefully for long enough for whatever your heating cycle is. At which point you need to cool it down again equally slowly. I think that's the only viable strategy for normal temperature electronics which doesn't involve active cooling.

Edit: normal batteries do not like high temperatures either, although there are special high-temperature molten-salt ones used in some missiles.

I am going to take an educated guess and suggest that @brahns is probably right and it's related to the process used at your contract manufacturer. Some parts of microcontroller boards (the crystal in particular and sometimes the reset circuitry) are sensitive to leakage on the board and the use of "no clean" flux can totally screw things up.

There seems to be contamination left on the board that can be affected by moisture humidity and cause excessive leakage. In the case of sensitive instrument boards, the slightest bit can cause problems. Traditional RA fluxes require solvent cleaning (which is a pain for the contract manufacturer, lots of environmental issues) but they're the best. I think water soluble is somewhere between the two.

The contamination is extremely hard to remove- solvent and mechanical scrubbing does it, but it's not easy nor is it cheap.

Anyway, that would be the first thing I would investigate, and see if they've made any changes to their process recently (of course they may be cagey about it, for fear of lawsuits or other unpleasant repercussions, so you may want to go at this laterally- compliment them on how nice and shiny the boards are or something- did you change anything?, then whack them with the 'we have a problem').