If I get enough wrong that I lose my status in that field, then I pick a few non-payoff ones to answer to push me back over the line: then I go back to ignoring the non-payoff ones again.

There is a problem there, but I don't think it's the one you describe above.

What seems to be going on here is that you find the non-payoff questions to be on average easier than the payoff questions. That's a problem if the reputation systems for the two are the same.

Lack of motivation to answer non-payoff questions isn't really a problem because - as you have proved - a player who believes themselves to be underrated will always want to answer questions, paid or otherwise.