Is this the current relationship between invoices and clients?
Clients::ClientID = Invoices::ClientID
Assuming that it is, in Manage | Database | relationships, make a new table occurrence of Invoices by clicking it and then clicking the duplicate button (2 green plus signs). You can doubleclick the new occurrence box to get a dialog to appear where you can rename the new occurrence box to InvoicesUnpaid.
We have not duplicated a table. Instead, this is a new reference to the same table already present in your database.
Add a calculation field, constNo, with "No" as it's sole term and select "text" as its return type. (I'm assuming "no" is the value entered into the radio button field when an invoice is unpaid.)
Now you can link this new occurrence of Invoices into your existing relationships like this:
Invoices::clientID = InvoicesUnpaid::ClientID AND
Invoices::constNo = InvoicesUnpaid::radiobuttonField
You can use this relationship to quickly check for unpaid invoices. One method is to add this text to your layout: "Client has Unpaid Invoices". Then use this conditional format expression: IsEmpty ( InvoicesUnpaid::clientID ) to change the font size of this message to 500 points. This makes the text disappear if there are no related records in InvoicesUnpaid. Now, each time you create a new invoice and select a client for that invoice, the message will appear automatically on your layout if the client has outstanding, unpaid invoices.