Is it permissible for a Muslim to register in official documents as being responsible for his mother and sisters in order to obtain a greater tax refund, and is the money recouped in this manner lawful, knowing that the father is their primary provider?

You are not permitted to state anything other than the truth to fraudulently obtain a high tax refund, because it is considered deception to take what you are not entitled to. The Prophet, peace and blessings be upon him, said: "Whoever cheats is not one of us, and scheming and deception are in the Hellfire." There is no harm in working for an entity that deducts legitimate tax with the hope of obtaining that refund, as long as the work is permissible, and the acquisition of the refund is done in a correct manner, without falsification or lying.