[Edit: The bug has been fixed in iOS 9.2, so the easiest solution is now "upgrade to iOS 9.2". I'm leaving the answer here in case it's useful to others for other reasons.]

Answering my own question, in case it's of any use to others: I found two possible solutions, neither of them entirely satisfactory...

1) Use the LR/Mogrify2 plugin from http://www.photographers-toolbox.com/products/lrmogrify2.php. My existing workflow used the Collection Publisher from http://regex.info/blog/lightroom-goodies/collection-publisher to produce a 3:2 jpg (2048x1365) in a folder that then syncs with my iPad. Using LR/Mogrify2, I added one more step, inserting a 4:3 (2048x1536) black background canvas. This is easy, and only requires a simple one-time settings change. The Photos app uses a black background for slideshows, so the black bars don't show when I'm flicking through an album in full-screen mode. So that's the solution I went for.

2) Use a different app instead of Photos. None of the apps I tried could work directly with the camera roll on the iPad; they all wanted to either upload the photos to the cloud, or make a second copy in their own storage. There were some good free cloud apps (Flickr, Carousel) and some local editors looked promising but expensive (Lightroom for iPad, obviously, and also Photosmith), but they all required a completely new workflow and/or constant internet access.