Hi biker
As far as I know, you don't need to buy the licence (vignette) for the bike if you are just on vacation here, and even many people who live here don't keep theirs up to date; you need to buy a new one each year, but I see many bikes with only ones from a couple of years ago.

As for what equipment the bike officially needs, fenders/mudguards are definitely not required. I had assumed that reflectors were required, but recently saw a police bike that didn't have them (or fenders), and so I asked the cop about the reflectors - he said that it is only the law to have them if you are riding at night (this was in Canton Vaud). However, when bike touring I always go with full fenders, reflectors, and a set of lights. The latter two are not only necessary at night, but are essential on Swiss roads because you can often unexpectedly come upon a tunnel, even in the non-mountainous regions. If you plan to follow any of the Swiss national bike routes then a bell is pretty essential as you may find yourself on some pedestrians & cyclists shared paths, which can be pretty full of pedestrians at times, and so a bell is very handy.

As for cycling over the Brunig pass, my wife and I did that this summer on a weekend trip from Lausanne to Luzern, and we posted photos here. To avoid going through the tunnel on the south side of the pass (which is about 500m long, I think), go through the town of Brienzwiler - there should be signs directing cyclists to take that direction. Your info regarding the road condition is correct - it is quite narrow and has a large volume of traffic on it. However, on the way up none of the large vehicles are going too fast, and on the way down you should be able to do a similar speed as the vehicles. In addition, Swiss drivers are generally reasonably courteous to cyclists and in a few thousand kms of riding here on all types of roads I've never had someone not give me the room that I needed. Even so, the Brunig Pass was not nearly as enjoyable as other passes that we've done because of having to pay attention to the constant stream of traffic. For half of the climb, a couple of mountain bikers rode up with us, and that was good because it made us feel a bit safer by being in a larger group. Quite a lot of cyclists do the route, so you may be able to wait on the pass for other cyclists to come by that you can go with (particularly if you do it during a weekend). Anyway, have fun!