Just a few things....
I happen to love Basel. It is a gem, with a great sense of humour.
A cable ferry (unpowered, just uses the power of the river) across the Rhine is plenty fun.
I think that three nights in Mürren is about right - it is important not to let a day of bad weather spoil your enjoyment of the mountains and mountain weather is notoriously fickle, if it were me I'd have 4 there but I love to walk, hike and explore. Years and years of going back and I'm never satisfied.
When you go from Mürren to Luzern take the cog wheel narrow gauge train ride from Interlaken to Luzern via the Brünig Pass. It is extremely scenic. Sit on the right along the lake. You don't need the fancy tourist only Golden Pass train, the little local trains every hour go on the same route, have the same map printed on the tables, and have modern air conditioned cars with panoramic glass all the way up into the roof, and actually have local people on them.
If you don't ask to go via Brunig you will probably be routed by the flatlands, not nearly as scenic.
If you do as I suggest, your train will turn around after a little while in Meiringen - where meringue was invented - and the location of the fight to the death between Sherlock Holmes and Professor Moriarty took place on Reichenbach Falls. Those real falls are just a few hundred metres from the station, and a funicular goes about half of the way up the falls - if you had time there is a scenic hike the rest of the way up.
As the train turns around (actually it reverses so the front becomes the back and the back the front) change seats so you are facing forwards and on the left. As you climb up from the valley note the waterfalls, including the Giessbach, descending to Lake Brienz, and as you go through the woods you can still see them if you look backwards. Enjoy.
I like the mountains around Luzern, but for me they are small potatoes compared with those in the Lauterbrunnen Valley. I've never been up any of them, but I have ridden the train up Rigi. Very nice, quite foggy at the summit the day I went, but it is a smoother mountain and I like craggy. YMMV, of course. The lake at Luzern though is incredible. I happily while away a few hours on a lake steamer and I like the tiny tiny cablecar up to Wissiflue.
A car would be a hinderance. Take the train, like the Swiss do.