Our 'Mini' cabinets are 20" wide (so 18.5" control panel) and look like this:

So there's a demonstration of the kind of space you will have. If you're going for 3 buttons then you may as well go for 4 as the space requirement will be roughly the same, plus if running NES you could then use the two extra as turbo-fire buttons for A and B, or map START and SELECT to them as I do; handy for those games that used those buttons for important game functions, such as the 'uppercut' button in Punch-Out (mapped to START).
It is doubtful you will fit any kind of standard trackball on a panel that size, many 3" trackballs have around a 5.5-6" footprint (remember underneath the panel is where you need to consider with switch, wiring and catch space), though you may get away with some kind of mini-trackball like a thumb trackball or similar.