Turns out it is a regex problem. I believe it is a truism that most problems are either DNS or regex.
As discussed in the thread linked below (and the thread that one links to) an incorrect regex causes problems for URLs with numbers in them, or something like that. Changing /etc/ood/config/ood_portal.yml from
host_regex: ‘cnode-10-0-20.+’
to
host_regex: ‘cnode-10-0-20.+?’
fixed the problem, and RStudio loads.
For some reason I wasn’t getting errors in the Apache logs, which is why I was having so much trouble tacking it down.