Time for some math.
If you have a 4000W inverter and it runs at 90% efficiency it will draw 4000W/.9 = 4444.4W from the battery.
The highest current will be when the battery is at a low SOC and the voltage is low, so lets use 24V. That means the current will be
4444.4/24=185A. Brief surge currents could be much higher. ( Note: It works out to 139A for a 3000W inverter)
Edit: Corrected the voltages in the following sentence.
I would say a 200A BMS would be the minimum size for a 12V battery with a 4000W inverter and 150A for a 24V battery with a 3000W inverter.
Note: I am not a fan of the Daly BMSs. My current favorite is the JK BMS line. Some of the newer JBD BMSs look very good, but I do not have enough info on them to be able to recommend them yet.