Can one preserve cooked fish with salt?
Disclaimer - I'm not familiar with the recipe, and I also couldn't find much on it when I looked, so I just had some thoughts to possibly offer in the absence of anything more offical.

If the salting is done before cooking, it wouldn't depend on the presence of salt "not being washed away" while cooking, but on the changes in the fish that happened while in the salt - it might be used for a short time to quickly draw water out of the fish, so it is drier and firmer when cooking, or it might be left in the salt longer, actually stored in salt for a while, to cure (like salt beef or pork).  The difference might be identified by how dry (and salty) the finished product is, since something cured in salt will tend to be very dry, maybe tough, and jerky-like in consistency, since salt is very efficient in extracting water, and may need to be washed and rehydrated before subsequent cooking, while something softer in consistency was likely salted for a short period of time and will depend on the cooking to stay unspoiled.  Salt-cured meat usually needs to be prepared further before eating (it is an ingredient, not a ready-made food).

If the salting was done after cooking, you would get different tastes and textures - probably mellower, less raw, and not needing further preparation (though it can be, it is a food moreso than an ingredient), something like that.  It isn't unheard of to preserve meats after cooking - you might look at confits or potted meats, to get ideas for how it would work.  Confits are also often briefly cured in salt and aromatics before cooking and storage, so this might be a related technique (or at least give some ideas).  Of course, confits and potted meats are also cooked or stored in fats (specifically for the air- and water-tight qualities, which keep bacteria from finding the food), which may be quite different from your Pla Tuu - and possibly the reason it needs periodic reheating to keep it safe while it is being, transformed?, during the storage or curing process or whatever.