To further add to what Mike stated, the size of the BMS has nothing at all to do with the capacity of the battery. You could build a 900Ah battery or 100Ah battery with the same BMS. Remember, Ah is amp-hours which is the capacity. The BMS rating is for amps which is an amount of current. Two different units measuring two different things.
Your 250A BMS should allow you to use up to a 3000W inverter (3000W / 12V = 250A). Of course if you were to try to fully utilize the 3000W for a while, your 300Ah battery will be 100% used in just over an hour (300Ah / 250A = 1.2 hours).