[everforward]

Distribute it.

E.g. one idea would be to make some kind of browser extension that scrapes tweets when the user loads a page from Twitter.

You could make it free by offering users some kind of feature in exchange (post tweets later or some kind of analytics or something). You could make it cheaper than paying for the Twitter API by paying users to install the extension (e.g. you could give out 4,200 $10 Amazon gift cards a month for the price of the Twitter API).

You could also use one of those networks that let you use people's home IP's if it's IP rate-limited.

Less ethically, you could try to pay an owner of an already-popular browser extension to bundle in a Twitter scraper. I don't know what that costs, but at $42k/month for the real API, you could probably pay a pretty penny and have it still be cheaper.