Is it permissible to issue fictitious invoices to entities that request them, while informing the employee dealing with them that they are simulated and not real, in order to obtain the rightful wage for extra work?

It appears permissible to write the aforementioned invoice as long as it has been designated to secure payment for your work, and it does not lead to anything prohibited, such as taking unlawful money or deception. You have clarified its reality and taken responsibility for what results from it.