The FTO is a conductive and transparent material. While cupper sheet is conductive but opaque. DSSC needs a transparent and conductive material. Such properties are realized by the use of conductive oxide such as TO or FTO.
You can use ........ But it will reduce the absorption as Cu is not transparent. Generally 10% Cu deposition is considered......... However, you might check whether it is forming Schottky contact or not, if Schottky contact is formed at Cu interface it will significantly reduce the Voc of the cell.
For dye-sensitized or organic solar cells, it is suitable to use a transparent conductive oxide as you mentioned, however, there exist a possibility to use Molybdenum instead of copper as the former has a lower work function than the latter, therefore, if you use copper, a Schottky contact would be formed instead of a simple Ohmic contact (which is more convenient for carriers transport). You can run a simulation using 1D software (SCAPS for example) and check the effect of different metallic layers on the performances (photovoltaic parameters as Voc, Isc...) of your solar-cell, this can be done systematically.