I cant guess what brand you have its nothing Obvious Im Familiar with, & its different from the Sun Race one you linkeed to.

so, I'll agree, Bike Shop, take it off , with their tool.

5 speed new freewheels are available , get the remover for the one you will keep using.

Shimano has made a standard spline for freewheels adopted across Asia where all of them are made now..

the 2 notch ones had a habit of damaging tool and the notches in the F/W ,
so I'd suggest ones with a spline fitting.