Yes, I crossposted this on stackexchange. I also found a solution to my own problem:

To get the access I wanted, I just started over again with the docker build instructions listed on the stackexchange page:

https://mathematica.stackexchange.com/questions/134637/running-mathematica-on-docker

After following the "new" solution on this page, I got everything working just fine.

Incidentally, my original problem was coming from a ready-made version of the engine image on the docker hub. The stack-exchange solution entailed making one's own build, but it isn't difficult to do.