Other than Luzern, I have spent between 3 and 10 days total in each of the places you mention, on separate trips with hiking as the main goal. I love each of them and would be hard-pressed to pick a favorite.
While Brig makes a logical central "base" if you don't want to change hotels, I think you'd be missing out on a lot of village charm if you stay there the whole time. And you'll be spending a lot of time on trains to get to a different place each day, even though it is central to them all (except Luzern).
If you are willing to change hotels a bit instead of staying in one base, you could consider heading to Lauterbrunnen on the route via Lugano, Arth-Goldau, and Luzern. Stop at Luzern for a few hours to explore before continuning on to Lauterbrunnen. There are lockers for your luggage in the train station.
After 2 nights in Lauterbrunnen, which will give you a chance to explore the BO, head for the Valais region via Kandersteg. Again, break your journey for a few hours and store your luggage while you explore. If it is a nice day, the ride up to Oeschinensee on the chairlift (assuming it is operating in October) is wonderful. Email the hotel at the lake and see if they will be open for lunch in October while you are there.
www.oeschinensee.ch
After a half-day in Kandersteg, head on south into the Valais region. You could go to Bettmeralp for 1 night and still have 2 to spend in Zermatt (to increase your chances of seeing the Matterhorn). Or, if you want to save a night for Montreux, spend both nights in either Bettmeralp or Zermatt. I prefer Bettmeralp, especially for the hike to the glacier, but Zermatt offers some nice excursions by lift and good hiking too.
On the other hand, if you truly want one base for the whole time, with trips from there, I would find Kandersteg more appealing than Brig. It is closer to Lauterbrunnen, so your trips thee would be shorter, but a bit farther from Bettmeralp and Zermatt.