What is the ruling on carrying out private work, not in violation of state regulations, using company equipment in exchange for paying its rental cost, especially given that the official of the government entity offers this work, which has limited profit?

The employee is entrusted with what is under his hand, and it is not permissible for him to use work equipment for personal purposes except with the employer's permission; because that is a forbidden transgression in Islamic law, and one must repent from it. This act is not made permissible by it being minor, or the profit being small, or by paying indirect compensation.