not all BB adjustable cups use a pin spanner, some have flats that fit a standard wrench. Its either one or the other...its simple to tell which just by looking at the outside of the Adjustable cup.
+1 get a lockring wrench. If there's 2 specialty bike tools that you'll ever want its a lockring wrench and a chain link press.
When putting the bearings into the fixed cup (which will still be on the bike) i've had good success squeezing a load of grease into the fixed cup and inserting the Bearings one by one through the hole in the cup.
Hold the frame sideways with the fixed cup on the bottom and just pop the bearings in through the hole in the cup from the bottom. The grease is thick and will hold the bearings in place.
That grease you have should be fine