I can't help but think there's something you aren't telling us, otherwise they've made a colossal error.
Let's look at your absolute worst case scenario. They cut off your API, put you completely out business, and end your app. Let's look at what they are offering you: $250. The worst case scenario + $250. What, exactly, do you have to lose, by rejecting their offer? $250. Put another way: nothing. Absolutely, nothing.
What is their worst case scenario? That you, oh I don't know, post around to bunch of places like Hacker News that they basically strong-armed you mafia style to shut down. Word spreads through dev circles that they are basically screwing their devs and abusing the trust of the dev community. They have to do all kinds of PR damage control.
The entire point of an open API is to attract devs to help spread your brand. If they want to turn around and start screwing the very people who help them, they'll find that dev pool dry very, very quickly.
In other words, the position they've put you in, you have virtually nothing to lose, and they have quite a bit. I'd respond accordingly. Save all their emails. Start working on the "zillow.com screws devs" website. Name your price, and tell them for both your sake, they should accept.