Luxury-ish, Less Touristy Recommendations

For spectacular Alps views and hiking, I would head for Murren in the Berner Oberland for a few days stay at the Hotel Eiger Muerren or the Hotel Bellevue. Then follow up with a stay at the Grandhotel Giessbach off Lake Brienz before overnighting in Geneva prior to your flight home.
Milan to Interlaaken is a 3.5 hour train 🚂 ride. It’s about another hour up to Murren via train to Lauterbrunnen then 🚠 cable car to Grutschalp then a narrow gauge train to Murren. You return to Interlaaken to catch the one- hour ⛴ ferry on Lake Brienz to the Grandhotel Giessbach.

Happy Travels!

For a little more "off the beaten track" location, try Kandersteg. Easy to get to from Milan taking a train to Brig, then switching to the "Loetschberger" train. Lots of good hotels. Take the lift up to lake Oeschinen for hiking and fantastic scenery.

Continue on to the train to Spiez and then the train to Zweisimmen then to Montreux. Continue to Vevey on Lac Leman. Swiss vineyards line the slopes along the lake, so wines you don't find in America are available. Frequent trains direct to Geneva airport from there.

Save Switzerland for its own 2-week trip. Going there only for a few days will annoy you because of what you won't be able to see.

Davos in the Engadine has some pretty decent hotels, it is the home of the annual World Economic Forum. There's also a pretty fancy spa in the Engadine.

I'm sure that if your wallet stretches to it this area could tick the your boxes in your post.

You could take a look at the Gstaad and Schönried area. Some very nice hotels in Zermatt. Fondue and Fendant.

The Berner Oberland will be every bit as “touristy” as Zermatt ( actually more so in early June), and farther off your travel path if you wish to fly home from Geneva. (Which is a good idea, as Lac Leman, aka Lake Geneva, is at the center of the Vaud, the Swiss wine region.

https://www.winetourism.com/wine-region/vaud-wine-region/

Take a look at the rail map of Switzerland to see the possible options and routes. Find Domodossola in the lower middle, in the blue area of Italy. That is where you will head from Milan to travel to Switzerland.

https://cdn-doc.myswissalps.com/docs/default-source/rail-network-maps/swisstravelpass.pdf?sfvrsn=7788603a_56

Zermatt is a good choice for alpine splendor and luxury accommodations. Not all upscale hotels will take children, but this lovely hotel, Coeur des Alpes, welcomes them:

https://e.coeurdesalpes.ch/#about

I was going to suggest Riffelalp Resort above the village, but unfortunately they do not open until 23 June. Coeur des Alpes will open 2 June. It will still be “low season” then so not as many tourists around.

Have a look at Riffelalp for next time:

https://www.riffelalp.com/en/welcome/

you could take the train up to Gornergrat and walk back from there (for hiking, it will depend on the snow level, but if it permits you could take the train up to Gornergrat for the view, then partway back down, and hike to Stellisee for a famous view (you need a nice day for this). Will you have a backpack child carrier for your daughter?

https://www.zermatt.ch/en/Media/Attractions/Stellisee

If you have sufficient time, add a night or 2 somewhere in the vineyard area above Lac Leman. And spend your last night before flying out in a small town on the lake, such as Morges) instead of busy, business-like Geneva. From Morges it is a 40-minute train ride right into the Geneva airport.

Unfortunately, it appears the Grandhotel Giessbach does not accept young children as guests. As Lola stated, many luxury properties do not allow children.