WG/coax transition: probe power handling capability

Maybe for the power handling capabilit of the pobe as your title showed.

Probe of large dimension has large seperation between the inner and outter conductor indicating a higher breakdown voltage, thus higher power handing capability.

But they recommend probes that get even wider inside the waveguide. What is the reason for this?

That may be for a better matching. For a probe(air filled coaxial cable) the ratio of 2.303 between the outer conductor and inner conductor would result in a 50 ohm impedance.

But when the probe to excite the wave guide, in which only the inner conductor of the coaxial cable is in the wave guide, the separation between waveguide's side wall and inner conductor of probe is much larger than 2.303. The impedance of the probe may change. So, when enlarge the section of probe inside the waveguide, an impedance that is closer to 50 ohm could be achieved.