Cooperative positioning with multiple low earth orbit (LEO) satellites is promising in providing location-based services and enhancing satellite-terrestrial communication. However, positioning accuracy is greatly affected by inter-beam interference and satellite-terrestrial topology geometry. To select the best combination of satellites from visible ones and suppress inter-beam interference, this paper explores the utilization of flexible beam scheduling and beamforming of multi-beam LEO satellites that can adjust beam directions toward the same earth-fixed cell to send positioning signals simultaneously. By leveraging Cramér-Rao lower bound (CRLB) to characterize user Time Difference of Arrival (TDOA) positioning accuracy, the concerned problem is formulated, aiming at optimizing user positioning accuracy under beam scheduling and beam transmission power constraints. To deal with the mixed-integer-nonconvex problem, we decompose it into an inner beamforming design problem and an outer beam scheduling problem. For the former, we first prove the monotonic relationship between user positioning accuracy and its perceived signal-to-interference-plus-noise ratio (SINR) to reformulate the problem, and then semidefinite relaxation (SDR) is adopted for beamforming design. For the outer problem, a heuristic low-complexity beam scheduling scheme is proposed, whose core idea is to schedule users with lower channel correlation to mitigate inter-beam interference while seeking a proper satellite-terrestrial topology geometry. Simulation results verify the superior positioning performance of our proposed positioning-oriented beamforming and beam scheduling scheme, and it is shown that average user positioning accuracy is improved by $17.1\%$ and $55.9\%$ when the beam transmission power is 20 dBw, compared to conventional beamforming and beam scheduling schemes, respectively.