September, hands down.
We've (CA residents) been to Switzerland 28 times in past 30 years, basing in multiple different regions for the first 10 or so visits to explore where we really wanted to "land" for our annual trips there.
After all that enjoyable exploration we now always base in Wengen. Quintessential 'car-free' mountain village, with 2 wonderful grocery markets, a bakery, plenty of really good restaurants, lots to do from this base. Typically we stay 10 or more days. One year we based there for 3 months and never ran out of things to do and see. (Happened to be 2001, that was interesting)
With the caveat that no one can predict the weather months in advance (heck, in the Bernese Oberland best you can count on is 24 maybe 48 hours in advance) here are my thoughts:
May is too early, most of the nice scenic walks at elevation don't open until mid-June. May is mud season.
After mid-June you could be okay
However, in our three decades of annual visits we have totally settled on first half of September. Only once did the winter weather come in during early September. (Happened to be 9/11/2001 year, that was interesting) First half September the weather is typically the best of the year. A lot of the summer crowds have dispersed. I don't know where you are seeing brown in the alps in September but, in Wengen, all is still luxuriously green
How long will you be in country?