Under the REST API we use /orders?paid=true to determine if an order has it’s balance paid or not.
Under the XML-RPC API you could use the following command to get the list of ones that are paid:
<?xml version='1.0' encoding='UTF-8'?>
<methodCall>
<methodName>DataService.query</methodName>
<params>
<param>
<value><string>yourapikey</string></value>
</param>
<param>
<value><string>Invoice</string></value>
</param>
<param>
<value><int>1</int></value>
</param>
<param>
<value><int>1</int></value>
</param>
<param>
<value><struct>
<member><name>PayStatus</name>
<value><string>1</string></value>
</member>
</struct></value>
</param>
<param>
<value><array>
<data>
<value><string>Id</string></value>
<value><string>ProductSold</string></value>
<value><string>ContactId</string></value>
<value><string>LastUpdated</string></value>
</data>
</array></value>
</param>
<param>
<value><string>LastUpdated</string></value>
</param>
<param>
<value><boolean>1</boolean></value>
</param>
</params>
</methodCall>