In this letter, we investigate balanced truncation for a class of continuous-time bilinear descriptor systems, appearing, e.g., in constraint circuit analysis or resulting from semi-discretization of the Navier-Stokes equations. For this, as widely done in the literature, we first aim at transforming the bilinear descriptor system into an equivalent ODE system by means of projectors for which the standard balanced truncation method for bilinear systems can be applied. Subsequently, we discuss how to solve the arising generalized Lyapunov equations corresponding to the equivalent ODE, without requiring the explicit computation of the projectors. The efficiency of the proposed algorithm is illustrated through a numerical example, which also shows its competitiveness with an iterative H ₂ -optimal model order reduction scheme.