Currently I have, starting at the wheel - a 5 foot diameter (ish) sprocket - cast iron, inch and a half pitch - with a special, irreplaceable chain on it. This drives a sprocket on a layshaft (2 1/2 inch diameter) in a moveable housing. This drives a six groove 800mm OD pulley - which currently drives a 132mm pulley on a small genny head. This head currently spins at 300ish rpm.
In order to spin a genny at 500+ rpm I need a 60mm or less pully - which is not possible on a 70mm shaft.. - so getting a step up gearbox will solve this.
OK, so let's work out all the sprocket/pulley ratios:
The wheel (60 inch sprocket) to 2.5 inch sprocket ratio is 24 (i.e., the 2.5 inch sprocket rotates 24 times as fast as wheel).
The 2.5 inch sprocket to 800mm (31.5 inch) pulley ratio is 0.0794 (i.e., the 800mm pulley rotates 0.0794 as fast as the 2.5 inch sprocket).
The existing 800mm pulley to existing 132mm pulley ratio is 6.06 (i.e., the 132mm pulley rotates 6.06 times as fast as the 800mm pulley).
To go from 300 RPM to 500 RPM at generator shaft, we need to increase the existing 800mm/132mm pulley ratio from 6.06 to 500/300 x 6.06 or to 10.1
Solving 800/X = 10.1 which results in X = 79.2mm. So if you replace the existing 132mm pulley with a 79.2mm pulley you will get 500 RPM at the generator. You could also replace the existing 800mm pulley with a larger size pulley (or replace both existing pullies with some combination of two new pullies) to achieve the 10.1 ratio.
However, you seem to believe that you need a 60mm or less pulley? You also stated that your wheel is rotating at 3 RPM and your existing generator shaft is rotating at 300 RPM?
Wheel RPM times 24 times 0.0794 times 6.06 should equal your current generator shaft RPM. If I use 3 RPM for the wheel speed, I would calculate your existing generator speed to be 34.6 RPM. If I use 300 RPM for existing generator speed, I would calculate your wheel speed to be 26 RPM. So I either suspect that my math is incorrect, I am not understanding how your sprocket/pulley arrangement works, the sprocket/pulley diameters you provided are incorrect, or the wheel/generator RPM estimates you provided are not accurate.