I kinda wonder if you are taking the wrong approach here and rather than looking at addons and integrations, take a look at your lovelace config instead.
Given that HA itself is up, along with all your other integrations, what do you have in terms of custom frontend cards/scripts?
My thought is that perhaps there is something in the frontend that is blocking or timing out the websocket connection. When this happens, have you looked at the browser console to see what errors might be there? I know the first instinct is usually to check the server, but your issues seem to be client based rather than server based.