The syntax looks correct for placing the text "Take the Tour" into the sixth repetition of LineItemProduct. Without seeing the entire script and not knowing the basic design of your layout table, relationship etc, I can only point out general issues that might be a factor:
LineItemProduct might be the wrong field type such as a calculation a field that is not defined to have at least 6 repetitions, or perhaps is a number field.
The table occurrence name: "TourBilling" might not be the correct table occurrence reference for the context in which your script executes. You mentioned "self joins" and Go to Related Records, neither of which should be needed to modify a repeating field that is currently visible on your layout unless "TourBilling" is not the occurrence selected in Layout Setup | Show Records from.
And using repeating fields for line items is not the best design option for an invoicing system. A related table of line items with a portal to that table located on your invoices layout is much more flexible and easier to work with.