I’m not really sure what else could be wrong. Everything else on your Home Assistant docker install looks correct to me. According to the integration page the analytics show zwave.me is only used by 12 active installations, so there aren’t a lot of others using this unfortunately so a problem in the integration itself might not be getting picked up, or a fix might not get prioritized. It seems something is closing after a period of time (probably the websocket) and it can’t re-establish itself. That github issue links to a change with a python package that might be responsible, but as you mentioned there, Home Assistant container should have all packages and dependencies in there and up to date. If it doesn’t, then there is a problem with the docker build for Home Assistant container where it is not using the current/correct package.
I have not used zwave.me, but how is your zwave.me installed? What hardware/os is it running on? Is it the same machine?
Is there any reason you don’t want to switch over to the more popular and better supported zwavejs2mqtt option? That’s what I use and it runs in docker, and connects to home assistant by websocket just fine (you don’t have to use the mqtt part). I have not had issues with it and closing websockets.