In this work, we present a control framework to effectively maneuver wheelchairs with a dynamically stable mobile manipulator. Wheelchairs are a type of nonholonomic cart system, maneuvering such systems with mobile manipulators (MM) is challenging mostly due to the following reasons: 1) These systems feature nonholonomic constraints and considerably varying inertial parameters that require online identification and adaptation. 2) These systems are widely used in human-centered environments, which demand the MM to operate in potentially crowded spaces while ensuring compliance for safe physical human-robot interaction (pHRI). We propose a control framework that plans whole-body motion based on quasi-static analysis to maneuver heavy nonholonomic carts while maintaining overall compliance. We validated our approach experimentally by maneuvering a wheelchair with a bimanual mobile manipulator, the CMU ballbot. The experiments demonstrate the proposed framework is able to track desired wheelchair velocity with loads varying from 11.8 kg to 79.4 kg at a maximum linear velocity of 0.45 m/s and angular velocity of 0.3 rad/s. Furthermore, we verified that the proposed method can generate human-like motion smoothness of the wheelchair while ensuring safe interactions with the environment.