I am out of suggestions on how to eliminate the clicks when the frequency changes.
I still believe that it is coming from the Si5351 when the LO is being changed. That could be in the runs from either of the I2C control lines, the lines to the Si5351, or the lines to the Nextion. There is a lot of updates going on with the Nextion screen, so I doubt that it is that set of lines. The Si5351 is changed only when there is a change in frequency or USB/LSB. That seems to be more in line with what you are describing.
Let us know what you find.