I have also been searching for an answer to this questions for weeks.  Seems like such a basic thing... the closest I came is the Customer Balance Detail (you mentioned this report is available in Quickbooks Desktop) If you open this report and set the filter for A/R Paid to All, it will show all of the transactions for a customer for the requested date range.  Unfortunately it does not display a running balance but it does give a total current balance.  Hopefully this helps you a little.