A multi-cell lithium battery needs both voltage protection and charge balancing.
That board does not have balance capability; the typical effect is the battery stops charging as soon as the first cell reaches the cutoff point, regardless of the charge in the other cells.
I'd not be too concerned by the overall BMS current ratings - those are last-resort fault protection and the working conditions should be defined by the charger in use and a fuse for discharge limiting.
The important thing is the protection board rating is higher than the working charge and discharge currents & those are less than the cell ratings.
What you do not want is a false trip when the battery is within safe range.
If your load current will be well under 8A, one of these may be suitable - they are available from many sellers on ebay:
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Whatever you use, you MUST also use a charger with a regulated output voltage and an appropriate current limit - that is a separate thing again from the cell protection stuff!
If the BMS protection starts limiting things (other than for balancing) the cells are being pushed beyond their proper limits.