I would recommend having the fuse in the circuit. However, if very careful, and double-triple checking the battery polarity, you can charge the batteries without the fuse. The charging current is limited by the battery charger chip on the power supply. The fuse is there to prevent the batteries from melting in the case of a catastrophic failure of the system or connecting the batteries backwards. There is a big high current diode on the power supply board that is there to blow the fuse if the batteries are connected backwards.
The fuse is a standard AGC15 amp fuse. These are available at electronic parts dealers, Lowe’s, Home Depot, and auto parts dealers.