Speeding up VBA Code that Sets Pivot Table Filters
I find that when you set PivotTable.ManualUpdate = True, Excel seems to often set that back to False for many of the changes you can make to PivotTables.  It does depend on the change, but I think changing PivotFields is one of those things that triggers this.

So you could try going to every PivotField change and resetting ManualUpdate to True after it.  Such as this:

With Sheet5.PivotTables("PivotTable21")
    .ManualUpdate = True
    .PivotFields("Division2").CurrentPage = DivRef
    .ManualUpdate = True
    .PivotFields("Region2").CurrentPage = RegRef
    .ManualUpdate = True
    .PivotFields("District2").CurrentPage = DistRef
    .ManualUpdate = False
End With


I know it seems silly, but I've seen this work.