Warm clothing needed 2nd week in June?

Puffy jacket and raincoat, plus at least one light long-sleeve shirt, for sun protection if nothing else. No beanie or warm scarf.

I did not pack my puffer coat last June and I regretted it. It can be very cool in morning, even at the lower elevation of murren and wengen.

Like sasha stated, I would bring at least one long-sleeve shirt. I have found that I can wear a wool shirt multiple times before it needs to be washed.

From a Chicago cold weather perspective you only need warm clothing when you visit the highest elevations like mountain tops (Schilthorn, Jungfrau etc. ) early where you are not moving and just admiring the views.
In 4 trips to Switzerland (2 in May and 2 in July) I have never brought a puffer jacket or scarves (it has not tend to be windy with a chill like here in Chicago) or beanies; I generally layer underneath (two long sleeves [flannel and long sleeve t-shirt] or a long sleeve under a short sleeve) my springer/summer water resistant jacket and wear a baseball hat to protect against the sun if I plan to go to higher elevations. Everyone has different tolerances.

I think it depends on how cold you get and how you time your excursions with weather. It's the high mountains, and you are almost as far north as Seattle. It could be cold up high. And or wet. Those mountains, particularly in the BO, are pretty active weather-wise.

I'm also a light packer, and I spend time in the mountains when I'm in Europe and at home. I probably would not pack a beanie. Even in expensive Grindelwald and Zermatt I'm sure you could find a serviceable beanie for 20 bucks. Not a name brand mountaineering beanie, but something that'll be fine and keep your head warm.

Maybe the same for a long sleeve technical undershirt, although I would be more wary of buying in Switzerland as that might be a little more expensive. But again I probably would not carry one the whole trip, and if I found that I was wanting in Switzerland, I'd pick up something unimpressive but workable on site.

Scarves I think are personal. And personally I'm not a scarf person. But people who like them seem to love them, and see them as multifunctional for warmth, dressing up, swaddling any discovered orphaned infants, impersonating a babushka to evade police, etc etc. If you want to bring a scarf I think you probably already know :)

What I'm curious about is are you going to pack a rain shell? That's a piece that I never travel without. Over your puffer it would provide a touch of extra insulation and wind protection, and the hood over a ball cap or whatever keeps your head a bit warmer. I would prioritize a light shell way ahead of a beanie and an undershirt.

Also, there are puffers and then there are puffers. I'm assuming you're talking about bringing something in the Patagonia Nanopuff/Northface Thermoball level? Lower end of mid-warmth, light, synthetic filled? If you are going more ultralight and less warm than that level I'd consider some layers.

Anyway that's what I would do. You though? Cold is kind of personal. I'm happy enough to keep moving when it's cold, tolerate not always being comfortably warm, etc. My wife though gets shivering easily and always packs warmer as a precaution. My attitude is don't pack it if you only need it for a short time and can buy it. I'll pay 40 bucks to not have that stuff in my bag the whole trip.

Anytime you are in the mountains, anywhere in the world, you should anticipate large temperature swings in the course of a day. And a big change will feel cold even if it is relatively not.

One thing you could do it check the weather daily starting now on a website/app like Meteoswiss to give you an idea of what it is like currently.
Definitely bring a waterproof layer, and if your puffer jacket fits under this, then I think you are set.