LLM-based MAS are gaining popularity due to their potential for collaborative problem-solving enhanced by advances in natural language comprehension, reasoning, and planning. Research in Theory of Mind (ToM) and Belief-Desire-Intention (BDI) models has the potential to further improve the agent's interaction and decision-making in such systems. However, collaborative intelligence in dynamic worlds remains difficult to accomplish since LLM performance in multi-agent worlds is extremely variable. Simply adding cognitive mechanisms like ToM and internal beliefs does not automatically result in improved coordination. The interplay between these mechanisms, particularly in relation to formal logic verification, remains largely underexplored in different LLMs. This work investigates: How do internal belief mechanisms, including symbolic solvers and Theory of Mind, influence collaborative decision-making in LLM-based multi-agent systems, and how does the interplay of those components influence system accuracy? We introduce a novel multi-agent architecture integrating ToM, BDI-style internal beliefs, and symbolic solvers for logical verification. We evaluate this architecture in a resource allocation problem with various LLMs and find an intricate interaction between LLM capabilities, cognitive mechanisms, and performance. This work contributes to the area of AI by proposing a novel multi-agent system with ToM, internal beliefs, and symbolic solvers for augmenting collaborative intelligence in multi-agent systems and evaluating its performance under different LLM settings.