To properly ream it, you will need to use a drill that is about 0.025-0.050" undersize. Reamers are not made to move a lot of material, and the drill will most likely drill oversize. Ideally you want to ream about about 0.005-0.020" off the finished size.
With that said, for your application, you could also just drill it slow and oily with a 5/8" drill bit.
If you don't want to buy tools you'll probably never use again, I have whatever size drills and reamers you might need, so if you want to go that route, shoot me a PM.
