😃 FIXED IT!!!
I did a quick test to see if my host had the mysqli support, but apparently not.
I then wasted all afternoon trying to find what was wrong. Looking at the phpinfo() output on the command line as compared to a straight web page on the server seemed to indicate that the apache module wasn't reading php.ini properly. But despite all my digging and poking, I only succeded in making things worse, and having to hastily backtrack.
Finally, wondering if I'd set it up wrongly in the first place, I found a set of instructions on the PHP website.
Setting up PHP in CGI mode rather than as an Apache module did the trick. It was a complete shot in the dark, and as the instructions warn, this isn't a good idea on live web servers, but that isn't an issue for me as I'm only using this server for testing on my internal network. And if it was a live server, you'd be able to use the mysqli functions and load PHP as a module.
So, thread resolved. Thanks for your suggestions😉