Integrated heterogeneous service provisioning (IHSP) is a promising paradigm that is designed to concurrently support a variety of heterogeneous services, extending beyond sensing and communication to meet the diverse needs of emerging applications. However, a primary challenge of IHSP is addressing the conflicts between multiple competing service demands under constrained resources. In this paper, we overcome this challenge by the joint use of two novel elastic design strategies: compromised service value assessment and flexible multi-dimensional resource multiplexing. Consequently, we propose a value-prioritized elastic multi-dimensional multiple access (MDMA) mechanism for IHSP systems. First, we modify the Value-of-Service (VoS) metric by incorporating elastic parameters to characterize user-specific tolerance and compromise in response to various performance degradations under constrained resources. This VoS metric serves as the foundation for prioritizing services and enabling effective fairness service scheduling among concurrent competing demands. Next, we adapt the MDMA to elastically multiplex services using appropriate multiple access schemes across different resource domains. This protocol leverages user-specific interference tolerances and cancellation capabilities across different domains to reduce resource-demanding conflicts and co-channel interference within the same domain. Then, we maximize the system's VoS by jointly optimizing MDMA design and power allocation. Since this problem is non-convex, we propose a monotonic optimization-assisted dynamic programming (MODP) algorithm to obtain its optimal solution. Additionally, we develop the VoS-prioritized successive convex approximation (SCA) algorithm to efficiently find its suboptimal solution. Finally, simulations are presented to validate the effectiveness of the proposed designs.