DB,
As you're planning to tour both sides of the valley, basing in Lauterbrunnen makes sense as it's the transportation "hub" for that area. You probably won't be spending that much time there anyway, aside from sleeping.
"We do not plan on hiking anything at higher elevations, just simple hikes and trails (maybe with some elevation changes though)"
As you don't appear to be planning any difficult technical hikes, you could probably manage just fine with a low cut (unless of course you need the ankle support). My preferred travel footwear for many years has been the ECCO Trac II Low. It's designed for that sort of activity and has robust construction. At my age, my hiking activities could probably be categorized as "light" so it works fine for me.