In ultra-dense LEO satellite networks, conventional communication-oriented beam pattern design cannot provide multiple favorable signals from different satellites simultaneously, and thus leads to poor positioning performance. To tackle this issue, in this paper, we propose a novel cooperative beam hopping (BH) framework to adaptively tune beam layouts suitable for multi-satellite coordinated positioning. On this basis, a joint user association, BH design and power allocation optimization problem is formulated to minimize average Cram\'er-Rao lower bound (CRLB). An efficient flexible BH control algorithm (FBHCA) is then proposed to solve the problem. Finally, a thorough experimental platform is built following the Third Generation Partnership Project (3GPP) defined non-terrestrial network (NTN) simulation parameters to validate the performance gain of the devised algorithm. The numerical results demonstrate that FBHCA can significantly improve CRLB performance over the benchmark scheme.