Crypto currency mining could be the solution. If one request to the site generates more revenue than it costs in resources, the bot problem is solved. The author says that he is getting 15 bot requests to his site per second. That is about 36 million requests per month. How much does it cost to serve those? $1000 would seem high. $1000/36M = $0.00003 per request. How long would a crypto currency, that is suitable for…
It might be enough for the request to require more resources from the requestor than from the server, even if it doesn't actually give the server any money. I mean the requestor probably isn't going to be willing to dedicate more hardware "horsepower" to taking your search engine down than you are to keeping it up. That was the idea behind Hashcash.
As for coins, the current Bitcoin hashrate is about 200 exahashes per second, down from a high of over 250 a couple of months ago, and the block reward is 6.25 BTC until probably June 02024. At a price of US$24000/BTC that's US$150k per block (plus a much smaller amount in transaction fees) or about US$1.25e-18 per hash. So your suggestion of US$3e-5 would require about 2e13 hashes. https://en.bitcoin.it/wiki/Non-specialized_hardware_comparis... says an overclocked ATI Radeon HD 6990 can do about 800 megahashes per second (8e8) so you're looking at about 3e4 seconds of compute on that card, about 8 hours.
Maybe one of the altcoins that uses a hash function with a smaller ASIC speedup would be a better fit, although I don't know enough about mining to know if there are any where GPUs are still competitive. Still, it seems like it might be more than a few seconds?