The best method is PCNL and incision of diverticul orifice or dilation.

The most important issue in my opinion is to increase the communication between the diverticulum and the urinary tract (calix or renal pelve). Otherwise, in future another stone can occur based on the stasis of urine in the diverticulum. You can do it by flexible ureterorenoscopy or percutaneous access, depending on your expertise.

I am afraid that there won't be the best solution for this particular condition. A variety of  situations will be met. Fortunately, most diverticula are located in the upper pole. So RIRS combined with or without percutaneous puncture can be used. Usually I do supine PCNL with ultrasound and fluoroscopy. Some in the middle -pole  can be taken care in this way as well. The most difficult is perhaps those in the ventral side of the lower pole. The handling could be only  relied on the PCNL. Michael Grasso from New York Univ. introduced his way of using furs to extract the guide wire from the diverticulum for safer tract establishment. Some how I had just 1 case handled with PCNL only.