Its not above 800C the entire time.
On a diesel, where water injection is proven to work, combustion tempatures can hit 3,000F+ at the peak but exhaust gas temperatures drop to 400-800F during cruise.
Real engineers dont use ideal gas for steam.
After me and my engineer buddies got lost in steam tables, ideal gas law, studying the lubercation properties of steam/water to figure out why water injection boosts fuel economy and power of diesels and gas turbines we setteled on this:
Water injection - EcoModder
Remember this is for a diesel, I am still convinced that for about 95% of the time when you try to run water injection continiously during cruise on a factory production (Original Equipment) gas engine you are going to hurt fuel economy.
Its going to take a lot of work, lots of testing and lots of tuning to produce a positive result on OE gas burners. But you can try.
I bet I can do it, but don't have a way to test it, yet.