+xAnd wouldn't you know, a minute after posting, I figured it out.
The filenames for the picture files in Inquisit (i.e. the contents of the items attribute for the picture elements) contained captial letters. I replaced those with case letters and now the picture displays
Names of any files are converted to all lower case letters during upload, this is done for cross-platform compatibility purposes (*nix file systems are case-sensitive, Windows file systems are not). For stimulus elements (e.g. <picture>) Inquisit itself fully controls, it doesn't normally matter whether you've given file names in the script in all lowercase, uppercase or mixed case, it will convert the file name to all lowercase internally and automatically. HTML content is a little different, since Inquisit doesn't fully control that and URLs themselves are case-sensitive. I.e. example.com/img/a.jpg and example.com/img/A.jpg are not equivalent. Here, then, it becomes important to have file names in the script given in all lowercase, or -- if you want to get fancy -- apply tolower() in the HTML source.
<img src="<%tolower(picture.BobFace.item(1))%>" style="width:15%">
(As an aside: I've developed the habit of always using strictly lowercase file names from the get-go as a "defensive programming" measure, so I don't ever have to think about or worry about things like this in scripts I write. For fixing up or revising scripts from other sources and long item lists in upper or mixed case, converters like https://convertcase.net/ are a handy tool.)