Recently, a new linearized gravity model, Very Special Linear Gravity (VSLG), has been developed within the framework of Very Special Relativity (VSR). One of its peculiar feature is to mantain gauge invariance while introducing a mass $m_g$ for the gravitational degrees of freedom. Therefore, the consistency of VSLG has been put to test by calculating the gravitational radiation produced by a keplerian binary system, exploiting Effective Field Theory's techniques. The resulting energy loss and period decay rate show a continous limit to the General Relativity result when sending $m_g \to 0$. Finally, we were able to constrain the VSLG mass parameter by using astronomical data from binary pulsars. The obtained upperbound of $m_g\sim 10^{-21}eV$ is still slightly weaker than the kinematical one obtained by the GW170817 detection, which should still hold in VSLG.