We took this tour in June of 2019, so a few years ago. We had HOT weather all the way through (even in Switzerland). I think it was considered a heatwave, so maybe not a typical experience. I did fine with capris and short sleeved tops. I had a lightweight fleece/sweatshirt, but never really needed it, except maybe for the bus. I also had a light rain jacket/windbreaker, but I don’t think it ever rained.
When we went, there was basically a free day in Lauterbrunnen where you could do your own thing. We joined the guide’s optional excursion to the Schilthorn. It was slight chilly (early morning), but not uncomfortably so.
For us, we didn’t choose to do any serious hiking that required special shoes (but of course maybe you have more ambitious plans). We both had ordinary running shoes, and were fine. After the trip to the Schilthorn, my husband and I returned to the valley with a combination of walking between the villages (paved path, easy but downhill of course) and one cable car. We also walked along a nice path on the valley floor back to the hotel and it was flat and easy. I don’t think the trail runners would be a bad idea, though, depending on what your walking shoes are like. You might also use the trail runners at other times on the trip (there’s a lot of walking).
I suppose you might have cooler weather in Switzerland, maybe take a couple of things to layer (ex fleece + raincoat) that would be useful at other times. Or a winter knit hat and light gloves can make a difference in mountain weather and don’t take up much space.
It’s a great tour, enjoy!!