Have you checked the input DC voltage to the inverter when it cycles on and off?
Constant voltage means the inverter itself is cycling off and on. Could be due to an internal inverter problem, or the inverter itself is activating its own low-voltage disconnect due to low input voltage (typically <11v).
If the input voltage to the inverter cycles off and on, then the BMS (actually anything in-between the cells and the inverter input) is suspect. If the cells are in a low SOC, the BMS may be simply cycling between low voltage disconnect/cut-off (voltage sag with inverter load) and LVD release (voltage recovery after the inverter load is removed). If you fully charged the cells after the LVD, then it's obviously another issue.
Not having access to the BMS's parameters/faults leaves you with few options in terms of troubleshooting the BMS. Have you checked voltages of the individual cells with a voltmeter for any anomalies?