I was in Switzerland in 2015 during the first couple of weeks in June with my grandson, and we saved a trip to Wengen for the very last days we were there. We stayed the nights of the 14th and 15th.
The special trail from Männlichen to Kleine Scheidegg had opened a few days before that. We went up and found we could walk it, although there was a lot of snow still around. It was completely socked in. The micro-weather forecast had said that there might be a few hours break in the clouds. Did not happen. We went to Mürren for lunch- still socked in...and then to Brienz. No sun, but below the clouds at least.
The critical element- the lift from Wengen to Männlichen - was running.
That lift starts running sometime in late May or early June, depending on skiers trying for their last snow or optimistic hikers hoping for no snow.
You can use the timetable:
which will have explanations in the footnotes, and also check with he Wengen or Interlaken tourist offices as the days approach.
the stations to use are Wengen, Wengiboden and Männlichen LWM for the cableway.
The conditions change rapidly in the middle weeks of June, and depend on how much residual snow pack there is.
if the dates for your trip are fixed, I'd suggest that you push the Jungfrau region to the very last days.
I have never found a Swiss hotel that requires no change in reservations if booked directly, until you get somewhere nearer the actual date....3 months for the fussy ones, less for the less fussy.
And, if you book directly with the hotels, especially if you phone them having checked out rates on the internet or e-mail directly, you can get the best rates, because they don't have to pay booking fees to the booking services.
In that region they all speak English, and some might appreciate your pure and long established version of French if that is what you prefer.
You can of course use Brienz as a base, but the total of 12 nights close to the Jungfrau region will give you a chance for some good weather days, and perhaps you would want to aim for the last days of your schedule.
Plenty to do in the area:
You could profitably spend more time in the Lake Luzern area, which will be well into Spring at that lower altitude. Good for many days: