Is it possible, at least in isenthalpic processes, that the Wet Bulb Temperature (WBT) is independent of the relative humidity?

Yes, it is quite possible in ideal cases when, as you said, the process is isenthalpic. Wet Bulb Temperature (WBT) is the temperature of a parcel of air when it's completely saturated with moisture when the parcel is enthalpically forced to absorb moisture. The water in contact with the air needs to be evaporated to expedite the absorption phenomenon. Surrounding air provides heat for evaporation, and the resulting vapor gets added back to the air, energy conserved as a result Still the overall temperature, known as the familiar Dry Bulb Temperature (DBT) drops a little. If this process persists until the air becomes completely saturated, the DBT reaches a minimum value that is called the WBT. As can be seen in such ideal processes that happen to be isenthalpic, the relative humidity (moisture content in air) goes on increasing at a constant known value of the WBT, corresponding to the DBT when the air would eventually be saturated with moisture.