Formula for On Click field to change the value manually by typing that is display from a reference table.

My first thought is something like this, with numeric fields and Yes / No checkbox fields. The checkbox fields could have code similar to this:

if 'Edit Height' = true then
  Height := null
else
    if 'Edit Height' = false then
      Height := first(Cartons.Height)
    end
end

Only with numeric fields.

Another approach using a Choice field with "Trigger after upate" set to:

if text(Choice) = "Edit" then
   Height := null;
   Width := null;
   Length := null
else
   let c := text(Choice);
   let t := (select Cartons where Description = c);
   Height := first(t.Height);
   Width := first(t.Width);
   Length := first(t.Length)
end

Just to add some diversity...

Since the "Carton" table is linked, I suppose that you have, in the "Shipments" table, a n:1 reference field called, by default, "Carton". In its "Trigger after update" optional configuration field, you could put:

Height := Carton.Height;
Width := Carton.Width;
Length := Carton.Length

This way, after selecting a linked carton record, the numeric fields in the shipment depicting the dimensions are initialised, and you can then modify them if needed. If you feel that it could sometimes be necessary to reinitialise them to the defaults for the chosen carton, define a button with the same script.

I like Alain's suggestion best. It replaces the Choice field with a n:1 reference field, which acts similar to a pop-up with an unlimited number of items. Plus it requires less code. Thank you Alain.

Since the Length, Width and Heigth fields must be Number fields for this solution to work, they are indeed changeable. The role of the script is only to write the default values when a "Carton" record is chosen, but after that the fields are yours entirely. This is perhaps what you were trying to achieve before resorting to Function fields, which copy the reference values at all times, and are thus not editable.