In established or historical usage, to 'throw' a competition is to deliberately lose it, particularly when in a leading position. For example, an athlete might throw a race, when in first place, by deliberately slowing down to let the person behind pass and win. In this case, the character in the book has deliberately misspelled (misspelt in British English) a word to let someone else win.
As you already understand, a 'spelling bee' is a type of spelling contest popular mainly in the United States.
In sporting events, races, matches, etc, can be 'thrown', for example by players bribed by criminals who bet heavily on the opposing side or player.
throw verb (LOSE INTENTIONALLY)
[ T ] to intentionally lose a game,
etc., for example because you have been paid to lose:
He was accused of accepting a bribe to throw a game.
They suspected the team of throwing games for money.
Throw (Cambridge Dictionary)
In more recent times, especially in certain contexts or milieux (e.g. the gaming fraternity, people on Reddit, sports commentators or journalists, etc), 'throw' can also be used to mean merely 'lose', 'fail to win', 'give up'. This last can be a shortening of the established idiom 'throw in the towel', which refers to the practice of boxers boxers or their seconds throwing a towel or sponge into the ring as a token of defeat. This meaning is clearly excluded by the context given in the question.