I would like to make a payment in the region of $1,000 to a small business in the UK, without them knowing who it came from. I have a UK bank account (HSBC) but it does not include checks, for what it's worth. Are there any easy ways to make such a payment? Conceptually, I'm thinking along the lines of how I've made anonymous donations to 'GoFundMe' pages, but in this case I need to target a business, about which I know nothing other than their business name and address.

Generally speaking, source of funds is vital , to exclude terrorist funding and money laundering, I suggest being upfront about it,

The small business would certainly need to account for the source of the funds too.

Mail them cash

But then, how does GoFundMe work? I've made several anonymous donations through GFM. I don't care that GFM knows who I am, I just don't want the ultimate recipient to know. I guess, in the case of GFM, if the government really wanted to investigate a 'donation', they could find out the details from GFM.
Unfortunately, I don't have access to UK cash. I suppose I could go to a bank in the US and order 'foreign currency' (like I used to do before traveling, before ATMs were so convenient and cheap) but that would be a hassle and a risk (of the cash going astray in the mail).