Is it permissible to return the one thousand Riyals to the mother after the burial, or is it considered the price of the trousseau? And in the event that it is not permissible, can the amount be given to her with the intention of showing filial piety towards her?

You did well in solving the problem and pleasing your father without outright lying, and this is commendable wisdom. There is no blame upon you for giving your mother money without informing your father; combining filial piety towards your mother with avoiding what angers him.