If the charger is putting out 85ish watts, it's not going to consume double that power from the AC line, not unless it's getting hot enough to be a fire hazard or at least to burn one's fingers. The energy consumed not sent to the battery has to go somewhere, and that somewhere pretty much has to be heat.
It is possible that it could consume 150 VA of apparent power if the power factor is poor. A 300W inverter should be plenty in any case and not break the bank. I would not expect any problems with an MSW inverter for this application.