It sounds like the position of the bookmark is locked in some way so that the BB cannot be inserted there. Perhaps if you offset the insertion point by one character then there wouldn't be a need to keep resetting the bookmark location
Code:
Dim objTemplate As Template
Dim objBB5 As BuildingBlock
Dim BBName As String 'Building Block String
Dim BM1 As String 'Bookmark String
Dim Rng5 As Range
BM1 = "Bookmark1"
Set objTemplate = ActiveDocument.AttachedTemplate
Set Rng5 = ActiveDocument.Bookmarks(BM1).Range
Rng5.InsertAfter " " 'two spaces
Rng5.Start = Rng5.End + 1 'collapse the range to between those spaces
BBName = "One"
Set objBB5 = objTemplate.BuildingBlockTypes(wdTypeAutoText).Categories("General").BuildingBlocks(BBName)
objBB5.Insert Rng5, True 'INSERT Building BLOCK
An alternative is the redo the BuildingBlock to include a space before whatever comes first