I don't want to get over excited but I appear to have fixed this issue with a "Software repair" via Xperia Companion.
In chat with Sony support they said, in brief:
You can download this here: http://www-support-downloads.sonymobile.com/Software%20Downloads/Xperia%20Companion/XperiaCompanion.exe
Open Xperia Companion and go to Software Repair. At this point there will be some on screen steps for you to follow, please do not connect the handset until you are prompted to do so.
Just to be clear, I agree with other posters that this issue came after the 6.0.1 OTA update and an on-board factory reset (in fact two) did not resolve it. Using Xperia Companion appears to have cleared it up completely.
It's a full factory reset, so I had to reinstall everything, but very much worth it to have my Pebble back to its old self. Hope this helps someone else.