Here is a start and some things to think about.
In the Jungfrau region, the most scenic regions are car free. Once you to get to Lauterbrunnen, Grindelwald, or perhaps Stechelberg, and look up at the mountains around, you will need to use public transport to get to them.
The costs are very different for the different parts of the transport lines. In an oversimplified description - if you go from town to town, it is relatively cheap public transport, so to go from Lauterbrunnen to Wengen is cheap. Go further up to Kleine Scheidegg and it gets expensive; that is an excursion line.
This map can help you understand the lay of the land, and if you zoom in you can see the train and cableway lines:
map.search.ch/.en.html ?pos=634216,168272&z=64 &poi=bergbahn,halteste lle,parkhaus,zug
<<I have read that a lot of towns s are closed this time of year and the weather unpredictable.">>
The weather in the Alps is unpredictable at any time of the year. The Alps make their own weather. However, forecasts are good for about 24 hours or so. The weather comes in layers...it may be clear on high and cloudy in the valleys, for example. The attached image taken from Wengen shows the Lauterbrunnen Valley on a clear day.
The mountain villages or towns are not closed. Simply put, Winter is still hanging on, Spring does not come until mid-June at the higher altitudes, the snow is not good for skiing, and there are no customers, so many hotels and restaurants close in the mountain villages. Many stay open. Cableways tend to be closed for maintenance and open on about May 25. But, even if you could ride the cable car (die Luftseilbahn) up from Wengen to Männlichen, for example, you could not hike because the trail would be closed due to snow remnants and the chance of snow slides blocking some or most of the trails.
Interlaken will be busier, but it hasit is a busines and travel hub, not a really scenic destination.
Have you considered Luzern?
It is at lower altitude, some lake boats will be running, it will be warmer, and snow will likely be there on the nearby mountains, but the lower places will be snow free, probably.
The transport museum might interest your child..
Driving a car in Switzerland is not cheap if you go into the cities. The large parking garage at the main station costs about 50 CHF for a 24 hour stay, and most hotels ( expect the luxury ones) don't have car parks.
How will you drive from Italy? Many routes, but not all, will be blocked by snow until early or mid-June. Make sure you buy the Swiss windshield sticker (carnet) at the border to use the express highways, or else you will pay a healthy fine.
The timetable can help you understand some of the travel options in the Bernese Oberland near Interlaken
This link, though oriented toward trains travel, may give you some ideas what to explore near the Jungfrau:
Check for opening hours.