Is it permissible for an employee who receives a meager wage for his arduous work, and who works in an institution that manipulates financial support and lays off employees, to "take his right into his own hands" by manipulating invoices, especially since he is a foreigner and does not easily find work?

You are not permitted to cheat and deceive under the pretext of being wronged by employers. As long as you have agreed to the wage, you are entitled to no more. It is obligatory for you to repent to Allah and return what you have taken unjustly to its rightful owners. The doors of provision are wide, and whoever fears Allah – He will make for him a way out and provide for him from where he does not expect.