Berner Oberland in April

Go up the Schilthorn only after you see that the weather report that is going to be clear; usually the best visibility is early in the morning. Stay in Lauterbrunnen Valley for great scenery and to avoid heights. Murren and Gimmelwald are all at higher elevations only accessible by gondola or cable car.

Wengen is accessed by train, as is the Jungfraujoch.

If you have time and inclination to visit a castle, Castle Chillon, near Montreux is interesting. And the train from Montreux to Interlaken (the one through the valley) is beautiful. Most here will say that Interlaken is just touristy, and not worth staying at, but you could take a boat around Lake Thun or Lake Brienz from there. They are a beautiful boat ride. My wife and I will be doing that in late April, train from Monatreux, and staying in Brienz. There is a nice cog railway, Rothorn Bahn, in Brienz, but it won't be running until May 1st, I believe. A short bus ride from Brienz is the Ballenberg Open Air Museum, you could find it online and see if it would interest you.

I would bring snow boots/waterproof shoes. However, it sounds like your wife will me miserable. From the valley to Murren is a gondola on each end of the valley and a cliff side train between the two. Can your wife manage her fear if she looks the other way or closes her eyes? Or if you and your daughter go and see what everybody goes to Switzerland for will your wife be okay staying in the valley or in Interlaken? The big gondolas don't scare me, but I road up to Murren in the service gondola with the lid open, I sat on the floor. Beautiful but scary to me. J

your wife wouldn't enjoy the Schilthorn I'm afraid. Bus to cable car to cable car (there may be a third, I'm not sure) and then right up to the summit with a revolving restaurant. Sounds to me like exactly what she wouldn't want.