Is it permissible for an employee who did not receive a retroactive housing allowance, knowing that he is entitled to it, to compensate for that by increasing his overtime work to receive more pay than he actually worked?

The arbiter in a dispute between an employee and an employer is the employment contract, for the contract is the law between the contracting parties as long as it does not contravene the Book of Allah the Almighty. Therefore, the employment contract with the company should be examined. If it contains provisions that prove entitlement to a housing allowance, then you have the right to claim it. If they deny this right and you are able to take it in a way that involves no transgression or injustice, then there is no sin in that.