A complaint isn't usually the best way to resolve a "live" service issue, it works on completely different timescales to what's necessary in a loss-of-service scenario - I'd usually suggest saving that for the "tidy-up" phase after resolution.
Repeatedly chasing CS, and documenting every contact, really is the best way here. I'd also question how things are left each time - are CS figuratively shrugging their shoulders and leaving unanswered questions, for example? If so, politely refuse to terminate the call until next steps are clearly outlined or solutions offered.
Without knowing what CS are actually offering by means of advice in this scenario, a technical escalation through to - ultimately - a single user fault being raised, is likely the correct outcome. Has that been offered?
