Battery energy storage systems (BESS) have emerged as a critical component in maintaining power system stability through frequency regulation. Their rapid response and flexible characteristics have generated considerable interest among researchers. This study focuses on the provision of a fast frequency response service, known as Dynamic Containment Frequency Response (DCFR), in Great Britain (GB). It conducts a detailed assessment of BESS-based DCFR service for frequency regulation and State-of-charge (SOC) management, including the configuration constraints set out by the energy recovery rules and SOC management impact. A methodology is presented to investigate the performance of DCFR-based BESS in a power system, alongside a stability analysis focusing on the impact of the SOC management mechanism. The stability study investigates the potential influential factors of battery SOC management when providing DCFR via root locus. For simulation case studies, a power imbalance estimation method is utilized for gaining the input. Based on the stability analysis results, key BESS configuration parameters are examined in an integrated power system model: C-rate, SOC management range, ratio and target. Another influential factor, SOC management time delay, is also analyzed. Finally, a comparison between DCFR and the previous frequency regulation service, Enhanced Frequency Response (EFR), is conducted. The study reveals that improper SOC management in DCFR can lead to SOC oscillation, adversely affecting performance. However, with proper configuration, DCFR offers more favorable outcomes than EFR in terms of frequency quality, SOC levels, and battery degradation.