We investigate the combined occurrence of edge faults and vertex faults in the burnt pancake graph (\( BP_n \)). In this paper, we prove that \( BP_n - F \), where \( F \) includes pairs of end-vertices of matching edges and fault-tolerant edges, contains a Hamiltonian cycle when \( |F| \leq n-2 \) and a Hamiltonian path when \( |F| \leq n-3 \). This establishes that \( BP_n \) is \((n-2)\)-hybrid fault Hamiltonian and \((n-3)\)-hybrid fault Hamiltonian connected for \( n \geq 3 \). These results are demonstrated to be optimal under the given conditions, with all bounds shown to be tight.