Is it permissible for an engineer to repair the company's broken screens at home, and then inform the management that another engineer performed the repair, in order to save the high cost of repair at the dealership?

It is not permissible for you to take money for repairing the screen at home as long as the company has refused to allow you to repair it yourself. It is also not permissible to lie by claiming that another engineer will perform the repair. The fundamental principle is that it is not permissible to take the screen from the company to repair it at home without their permission, and you must adhere to the limits of your job. You are not justified in doing this by considering it to be in the company's interest from your point of view, for an agent only has the authority to act as explicitly or customarily permitted by their principal.