Yeah you can just ditch the cottered spindle and replace it with an appropriate square taper spindle and crankset (using the original bearing cups.) If you do this, do not bother removing the (driveside) fixed bottom bracket cup. Especially since you don't know which way to turn it.

If you already have a crank and BB to use, try it and see if it fits with the cups you already have in the bike. Otherwise, you will need a new square taper spindle and crankset.

+1 Measuring the BB shell will help. 68mm, 70mm, or 73mm?