What is the ruling of Islamic law on obtaining a false residency certificate for a relative of mine so that she may benefit from social housing, knowing that I fear if I refuse, she will cut ties with me?

Falsifying documents and data, or lying and committing fraud to bypass the conditions for obtaining housing, is not permissible. This is because the assessment of the benefit in such conditions falls within the jurisdiction of the authorities. Obedience to the paternal aunt and maintaining ties with her should be in what is good and permissible. If the paternal aunt severs ties with you for this reason, then she is the one at fault. You must be keen to maintain ties with her and show her kindness.