Constrained Coulomb gases arise in various domains of physics and mathematics, notably in statistical mechanics and random matrix theory, where particle interactions are governed by singular repulsive potentials under external confinement and additional constraints. Simulating such systems poses significant challenges due to the interplay between singular interactions and constraint conditions. In this work, we develop an efficient numerical framework based on adapted Hamiltonian Monte Carlo methods to sample constrained Coulomb gases accurately. We establish a theoretical foundation using the Gibbs conditioning principle, providing rigorous insight into the behavior of these systems under linear and quadratic constraints. Numerical experiments in one and two dimensions validate the theoretical predictions, demonstrating the effectiveness of our approach in capturing equilibrium measures and fluctuation properties. This methodology opens avenues for precise simulation of conditioned particle systems, with potential applications in physics, probability, and computational mathematics.