Josephson junctions are compact on-chip sources of coherent microwave and terahertz radiation, but conventional radiators couple the emission frequency, output power and dissipation through a single bias voltage. Here we demonstrate a gate-reconfigurable Josephson microwave radiator based on a graphene weak link integrated with a separately gated graphene shunt resistor. The bias voltage sets the Josephson frequency, while the local gates tune the Josephson current and the total device resistance, enabling electrostatic control of emitted power at a chosen frequency. The device resolves Josephson harmonics up to fourth order, and its radiation power scales with source current and impedance as expected from a lumped-element circuit model. We further show that the emission linewidth is quantitatively described by Johnson-Nyquist noise at the electron temperature. The linewidth therefore provides an on-chip thermometer under bias, with the resistance-dependent heating captured by a minimal heat-balance model based on Wiedemann-Franz diffusion and electron-phonon coupling.