All you need is a point sharp enough to create a continuous line of defects on the surface of the glass. The smaller the included angle, the less-robust the scribe will be -- but, hey, you clearly have a grinder to touch up the point.
The standard glass cutter is a little roller so it sort of has a fair amount of negative rake. And due to its small width it probably has a pretty small included angle. I think it works by just concentrating a lot of pressure on the glass so it fractures along the path you are cutting. So I'd suggest a grind that produces a pretty small included angle, and a generous amount of negative rake so the pressure from your tool fractures the glass, rather than scratching it.
The fixture you use to hold the bottle in place while you turn it probably is important in this situation, since your scribing tool can't rotate. The roller on a real glass cutter can't exert any tangential force on the glass, it's all perpendicular to the glass surface. You won't be able to achieve that with a ground HSS scriber.
An alternative would be to make something similar to a standard glass cutter, if you happen to have some air or oil hardening stock on hand. It wouldn't have to be all that thin, just drill a center hole through it, turn down a nice edge on the radius, harden it, hone it and mount it in something that permits it to spin.
I first came across mojitos in Cuba. Made with fresh-squeezed cane juice. So good!
