Outstanding! I'd given up hope of finding a solution, but with that
help, I think I've cracked it.
The preferred width, in combination with table.allowautofit, gives me
the settings I need.

Thank you, Tony.

In case you're interested, here's the function I'm using:

Function TablesAutofit(tbl As Table) As String
'AUTOFIT_WINDOW As String = "To Window"
'AUTOFIT_CONTENTS As String = "To Contents"
'AUTOFIT_FIXED As String = "Fixed Width"
'    - no autofit, all columns equal width
'AUTOFIT_DISABLED As String = "Disabled"
'    - no autofit, columns of variable width

' this function examines a table,
' determines preferred width type
' then checks preferred width
' if preferred width <100 and allowautofit = true: autofit to contents
' if preferred width = 100% and allowautofit = true, autofit to window
' if allowautofit = false and columns equal, fixed width.
' otherwise disabled.

    Dim TypePrefWidth As WdPreferredWidthType
    Dim i As Integer
    Dim FirstColWidth As Long, NewColWidth As Long
    Dim bAllColumnsEqual As Boolean

    TypePrefWidth = tbl.PreferredWidthType
    tbl.PreferredWidthType = wdPreferredWidthPercent
    With tbl
        If .PreferredWidth < 100 And .AllowAutoFit = True Then
            TablesAutofit = AUTOFIT_CONTENTS
        ElseIf .PreferredWidth >= 100 And .AllowAutoFit = True Then
            TablesAutofit = AUTOFIT_WINDOW
        Else
            bAllColumnsEqual = True
            If .Columns.Count > 1 Then
                FirstColWidth = .Columns(1).Width
                For i = 2 To .Columns.Count
                    NewColWidth = .Columns(i).Width
                    If NewColWidth <> FirstColWidth Then
                        bAllColumnsEqual = False
                        Exit For
                    End If
                Next i
            End If

            If bAllColumnsEqual = True Then
                TablesAutofit = AUTOFIT_FIXED
            Else
                TablesAutofit = AUTOFIT_DISABLED
            End If
        End If
    End With
    ' return table to the preferredwidthtype it started with
    tbl.PreferredWidthType = TypePrefWidth
End Function

Darren