Hi. I took both the My Way Alpine and the Best of Switzerland tours around this time of year. While there were some folks who did actual hiking in the Berner Oberland, I did more wandering on pathways. I had two pairs of hiking sandals - All Out Blaze Sieve by Merrell and Keen Newport H2. I don't recall excessive rain, but if either of these were to get wet, they would dry really quickly. I often take a pair of ballet type flats, too.
The weather can be really warm this time of year - in the 80s, so make sure you have typical layers including some short sleeve tops. I don't bring shorts because I don't have any viable shorts for wearing in public, but several folks did. And they wore them.
Essentials: a few pairs of pants - I wear knit black all purpose; some short sleeved tops; one or two long sleeved tops; a sweater; a waterproof windbreaker; something to sleep in (one of the tops?); 1-2 pairs socks; undies; hat. I usually have jewelry I wear everyday, with maybe a switch out of earrings to feel fancy if so inclined. I take one or more scarves/shawls. In summer, a pair of flip flops or similar may come in handy - for on the bus, in the hotel. I remember on one trip, not either of these, I was diverted to Frankfurt airport and had only rugged walking shoes with me.. Oh, how flip flops in my carry on could have eased my tired feet, but they cost an arm and a leg in the airport. And, my luggage was delayed, so I was stuck with the heavy clunkers for a few days. Lesson learned to provide an option of footwear in my carry on.
What was it you were thinking you might want to have shipped to you? Hiking poles? If so, google that and you will read what solutions others have come up with. Anything else, I think you can easily purchase - and it might be fun doing that. When you get a list of hotels, check if they have pools in case you wish to throw in a bathing suit.
One activity I wish I had known about: river rafting in Chamonix. On the last day of the tour, a bunch of us were sharing some wine on the tiny hotel lawn. The river ran at the bottom of the lawn, and lo and behold, a raft of laughing folks went by, followed by individuals who looked to be in wetsuits with some kind of individual inner tube device. It looked like so much fun.