To answer the question I asked about a year ago, the best storage backend driver appears to be overlay2 which was added in docker 1.12 and according to the docker page on overlay vs. overlay2, the “overlay2 driver addresses” the “known limitations with inode exhaustion and commit performance” of the “the overlay driver”.
It is however “only compatible with Linux kernel 4.0” but that isn’t a problem with my VPS as it uses a recent kernel (4.7 at the moment).
I didn’t really get around to working on my web site that much in the last year so the fact that it took about a year for this driver to come into existence didn’t end up being a problem.