There appears to be an undocumented value for the cmd attribute. In addition to Delete, New and Update, there is a Moderate value available.
When you need to update the "Approval Status" of a list item using the Lists.asmx web service, the following Method element can be used to set the Approval Status of the list item to Approved:
<Method ID='1' Cmd='Moderate'>
<Field Name='ID'>1</Field>
<Field Name='_ModerationStatus'>0</Field>
</Method>
If you try to update the _ModerationStatus with the Update Cmd, the call will succeed, but will NOT update the Approval Status field.