Reducing undesirable path crossings among trajectories of different robots is vital in multi-robot navigation missions, which not only reduces detours and conflict scenarios, but also enhances navigation efficiency and boosts productivity. Despite recent progress in multi-robot path-crossing-minimal (MPCM) navigation, the majority of approaches depend on the minimal squared-distance reassignment of suitable desired points to robots directly. However, if obstacles occupy the passing space, calculating the actual robot-point distances becomes complex or intractable, which may render the MPCM navigation in obstacle environments inefficient or even infeasible. In this paper, the concurrent-allocation task execution (CATE) algorithm is presented to address this problem (i.e., MPCM navigation in obstacle environments). First, the path-crossing-related elements in terms of (i) robot allocation, (ii) desired-point convergence, and (iii) collision and obstacle avoidance are encoded into integer and control barrier function (CBF) constraints. Then, the proposed constraints are used in an online constrained optimization framework, which implicitly yet effectively minimizes the possible path crossings and trajectory length in obstacle environments by minimizing the desired point allocation cost and slack variables in CBF constraints simultaneously. In this way, the MPCM navigation in obstacle environments can be achieved with flexible spatial orderings. Note that the feasibility of solutions and the asymptotic convergence property of the proposed CATE algorithm in obstacle environments are both guaranteed, and the calculation burden is also reduced by concurrently calculating the optimal allocation and the control input directly without the path planning process.