Effective sharpness depends on both the angle of the cone and the geometry of the very edge/point. In this case it sounds like you aren't aiming for very deep penetration into the surface so the tip of the point itself is likely to be the key factor.
You also need to consider the ways that a sharp point wears. In general this can either be that material from the point is worn away, chipped or that the point becomes bent. This is a combination of the mechanical properties of the steel and the geometry of the point. Most sharp tools are heat treated and the precise temper is a crucial variable in tandem with the geometry in determining performance as creating a close approximation of a geometrically 'ideal' point or edge relies on controlling a combination of abrasive wear and plastic deformation which is not well catered for by conventional tolerancing.
In practice specifying the manufacturing process is probably more meaningful than geometric tolerances in the context of sharpness. Normally sharp edges are finished b y abrasive grinding and/or lapping and it may be the case that a manual finishing process makes the most sense.
In terms of the scale and overall geometry you are talking about there is an obvious parallel with sharpening the tungsten electrodes used in TIG welding and it may even be worth considering these as an easily available source of hard, small diameter rod, High speed steel and tungsten carbide are also possible contenders.
It would probably also be worth considering having the larger diameter part as a separate holder with a collet or similar to hold a replaceable probe tip. This arrangement would certainly be easier to manufacture than turning a small diameter neck in stainless.
For example you might specify that the point is sharpened to say 60 degrees and finished on a 1200 grit abrasive wheel. Also bear in mind that if you just specify that you want the tip sharp then sharpening things is something that any machinist should immediately grasp.
It may also be that you can find an off the shelf source of points as there are quite a few applications where these are used, carbide tipped scribers immediately spring to mind.
Also as general advice if you are designing something out of your specific expertise and experience it is always worth talking to whoever is going to make it and telling them what you actually want the part to achieve.