As I suggested earlier, the spindle would benefit from being as large a diameter as the OP's application can accommodate. However, Belville washers are, IIRC, available in small diameters.
XD 351's comment about spring preload washers not controlling end-float is not strictly correct. They will not eliminate end-float, but can control it, to an extent that allows the design to be practicable. The force required to move the spindle axially can be designed to be higher than anticipated forces applied to the spindle by the cutter, etc. That this works in practice is borne out by the thousands of (fairly) satisfied Unimat users and its very widespread use in precision spindles.
The whole point of spring pre-loading in this application is because trying to set preload by screw-adjustment is difficult – you can't tell when it's correct and, in small sizes, the axial displacement required for reliably setting the correct adjustment is too small to be applied easily by a practicable-pitch thread, especially when a lock-nut disrupts the adjustment. If the adjustment of such a rigid set-up is such that the axial float is just eliminated, there will be minimal bearing preload – less than the 'correct' amount. However, setting the 'correct' amount is difficult – how do you measure it?
In much larger applications, opposed taper roller bearing-supported shafts are often set to a desired torque required to overcome the drag imposed by the preloaded bearing. Great fun, swapping shims, rebuilding, measuring torque with a spring balance pulling a string wound around a gear, taking it all apart and doing it all over again. And again. When, years ago, I rebuilt a Land Rover transmission, (LR service manuals detailed this method), I was surprised to learn that the local main dealer never opened transmissions – they just bunged in 'recon units'. The labour (and skill?) required was too much. Luckily, they carried boxes of shims for us idiots working for free, in our own time…
OK, enough of this counsel of perfection – don't be put off. You will probably get away with a screw-adjusted arrangement; make the thread as fine as you can and adjust it until there's just a little rotational drag, and there's no rattling on shaking the assembly. Spring pre-load makes life easier…
(Stupid emoji added to my previous post, apparently by 'auto-correcting' software built into this forum. Can't you just wait for self-correcting autonomous cars?)