From a financial reporting perspective (General Ledger and Financial Statements) where reporting is monthly, using any accounting date during the month will result in the same monthly revenue. Therefore if you use the actual invoice date as your accounting date, your Aging Report in AR should give you the results you want.  

The issue is when you run these reports by Accounting Date the partial cash receipts don't appear to be under correct "Aging" column

Hi John -

When running the AR Aging Detail reports by accounting date, the cash receipt will age using the invoice date.  If the invoice has a different date for the invoice and accounting date, the cash receipt will age in different columns.  If you run the report by invoice date, the cash receipt will line up however it will not reconcile to GL.  If you want to use the AR Aging Detail reports by accounting date and have your cash receipts line up with the invoices, then the invoice date and accounting date on the invoice need to be the same date.  See knowledgebase  article 26237, "How does a cash receipt age on the Aging Detail reports in Accounts Receivable?" for more detail on how cash receipts age on the detail reports.