Erroneous core update suggestion

This has been asked many times before, on this forum and elsewhere, but I can’t find an answer that applies to my situation, so I’m afraid I’ll have to ask for myself. It’s difficult to troubleshoot because I can’t make it happen on demand.

On one of my websites, periodically WordPress will start thinking that I need to update the core, even though it’s current (I even have automatic updates turned on). The little “updated needed” indicators at the top and left black menus will say “1”, and the following will appear at the top of every page: “WordPress 4.3.1 (or whatever is the current version) is available! Please update now.” The dashboard’s “At a Glance” area has the most absurd line:

WordPress 4.3.1 running CRASH child of Vantage theme. [Update to 4.3.1]

What?

If I then click on any of the numerous links to http://mysite.com/wp-admin/update-core.php, it will realize its mistake and get well: “You have the latest version of WordPress. Future security updates will be applied automatically. If you need to re-install version 4.3.1…yadayada…” Once I’ve gone to that page, the complaints at the top of the admin pages will also disappear, but after some time (a matter of hours, I think, or maybe days – it seems random) they will reappear. It has been doing this for quite a while, through several updates at least, possibly ever since I installed the site 4 months ago.

Yes, I did test deactivating all plugins and using the Twenty Fifteen theme – no change.

I decided to install Wordfence and run its scan to see if it can find any problems with the core code. It claimed exactly one “problem”:

Your WordPress version is out of date
Current WordPress Version: 4.3.1
New WordPress Version: 4.3.1

Argh! So, does anyone have any suggestions for what I can check?