I always remove the pump and pulley as a unit (which can be done without removing the radiator). Then separate the pump & pulley on the bench, with a puller as needed. (My opinion, it should be tight enough to need a puller. If not, use some Loctite.)
If you have the original bolt at the upper RH corner, the bolt stays trapped between the pump & pulley until the pulley is removed. But it can still come off the engine while trapped in place. I ran the aftermarket pump with the fixed pulley for many years, so I have a stud in place of that bolt.
With the studs at 2 or 3 locations, the nut cannot come off until the pump is pulled partway out. So, loosen each nut until it touches the pulley, then work the pump out to get enough room to remove the nuts. When reinstalling, start each nut by just a turn or three. When they are all installed, you can spin them down and do the final tightening.