It's October 1^st, and my server is still up and running [1]. No DDoS (Distributed Denial of Service) is happening as far as I can see. It appears that the “threat” was indeed, a joke. Not only did I not pay the danegeld, but I didn't have to move my site to Cloudflare [2].
And that Bitcoin address I was mailed? Still no funds in it [3] (I decided to make the Bitcoin address known—no reason to hide it), which makes me feel better about the whole thing.
[2] https://betanews.com/2017/09/27/cloudflare-unmetered-mitigation/
[3] https://blockchain.info/address/1W5FWQjvHGMkaPeuR4SvRNojsv64WgJNt