We propose a regularization method for nonlinear least-squares problems with equality constraints. Our approach is modeled after those of Arreckx and Orban (2018) and Dehghani et al. (2019), and applies a selective regularization scheme that may be viewed as a reformulation of an augmented Lagrangian. Our formulation avoids the occurrence of the operator \(A(x)^T A(x)\), where \(A\) is the Jacobian of the nonlinear residual, which typically contributes to the density and ill conditioning of subproblems. Under boundedness of the derivatives, we establish global convergence to a KKT point or a stationary point of an infeasibility measure. If second derivatives are Lipschitz continuous and a second-order sufficient condition is satisfied, we establish superlinear convergence without requiring a constraint qualification to hold. The convergence rate is determined by a Dennis-Moré-type condition. We describe our implementation in the Julia language, which supports multiple floating-point systems. We illustrate a simple progressive scheme to obtain solutions in quadruple precision. Because our approach is similar to applying an SQP method with an exact merit function on a related problem, we show that our implementation compares favorably to IPOPT in IEEE double precision.