There should be a hook for when the EntityView has finished rendering (code is in repo/resources/wikibase.ui.entityViewInit.js, add mw.hook(…).fire() shortly after the createEntityView call in L359). (Not everything might actually be done by this point – some things are asynchronous – but the main rendering should be good enough for now.)
We might want to add this hook to T169775: Declare Wikibase JavaScript hooks stable as well.