Obviously the first and most obvious way to verify he cannot win is to try and win and fail.In the general case it appears we are all in closer agreement than it would seem. On both the moral issue and the playstyle issue.
The points of disagreement are more about which cases this specific dragon encounter example falls under. It seems to me that there are a lot of details that could easily move it from one realm to the other.
For example, if the scenario isn't viewed as a no win scenario then just immediately acquiescing to the dragons request is very problematic. However if it is a no-win scenario where the only choices are either acquiesce to the request or you both die then the paladin chose the moral action.
With that said, there's one principle that hasn't really been discussed much in relation to the moral / not moral question. Moral decisions are always based on impartial knowledge. So the most important part of determining the morality of the paladins action isn't to look at it from the knowledge we have of the whole situation. Instead it's to look at the knowledge he had of the situation and what actions he could have taken to get more knowledge of the situation.
So in our example, the paladin is confronted with a dragon much to strong for him to fight and survive. The dragon gives him a verbal choice, "i'll kill you both unless you give me the NPC." At this point in the scene how can the paladin know that it's a no-win scenario? Is there any reasonable actions a person could do in this scenario to attempt to validate that it's a no-win scenario. Is it okay to give the NPC to the Dragon without attempting to validate it's a no win scenario first?
That to me is ultimate root cause of the paladins moral failure in this example, no attempt to validate that his situation really was a no win scenario before he gave the NPC to the dragon. It's also important to note that this moral failure exists whether it actually was a no win scenario or not and regardless of whether the paladin or his player believed it was a no-win scenario.
Most likely result, two dead.
keeps coming back to an requirement to keep trying in spite of previous efforts to win that failed?
not sure where that train ends.
if he tries to negotiate and that fails, then what? Find another "try a new way to win" to validate he cannot win? keep risking failure again and again and losing any chance of returning to right the wrong... asd infinutum?
or is there a number, some holy figure, minimum number of "risk it agains" after trying and failing before the oath is fulfilled?
Lets look at a similar example - same exact situiation with ONE change - he was carrying two injured victims to safety - a man and a child.
Dragon gives him the same offer, turn over the man and the rest go free.
Now, is it still beholden on the paladin to fulfill his oath to risk the child and himself to keep trying? Since we have established that this world ending quest wasn't enough to warrant the choice to give in to the dragon, clearly this extra child wont sway the oath-breaking repercussions.
After all, its got to be about the paladin and his oath... that is what heroic means, right?