You'll be fine with a 2mm or a 3mm cutter, obviously 3mm gives you a bit more tolerance for sawing so that's what I'd recommend for your first go at this technique. The price you pay is that flushing down the sawn edge is a slightly longer job.
Personally, I prefer to get a coat or two of finish on the entire box before sawing off the lid. But whenever you choose saw slowly, constantly adjusting the cut to remain central, and tape in small wedges to hold the kerf open and rigid.
The mating faces of the two sawn surfaces needs to be made absolutely flawless. Many people will judge the quality of the box by the quality of this shut line, so aim for perfection.
I often use one or a combination of two techniques.
First, if the grain is fair and the timber sweet working I'll hand plane the two mating surfaces flush. Your hand planing skills need to be pretty good to reliably pull this off. I find it easier with a slightly cambered iron,
Second, if planing will be problematic, then coat a 25mm thick board of MDF with some wide abrasive paper from an industrial sander. I think from memory mine has 120 grit on one side and 180 grit on the other side, by carefully moving the working around you'll guarantee two dead flat surfaces. It's worth putting some effort into making one these sanding boards as they last for years and are invaluable in all sorts of applications where you need absolutely precision components,
The end result should be a box lid that closes without any trace of binding and with a totally even shut line, fine quality work to this standard is fully within the scope of the home woodworker provide care is taken throughout,