I never heard the fan clutch click. you probably can find a u tube video with an example of one that does.
My water pump failed, sounded like a can of marbles, kind of. I heard the sound of a failing water pump first in the family Econoline van, just before it died when I drove it to a job interview when I was 17. what a mess and towing bill. mine failed while I was 500 miles from home, half way through vacation. I paid a local fella to change it out. I did the fan clutch a month or so later, but am not convinced there was really anything wrong with it. ( my ac is still flaky when it is around 100 and traffic is slow)
you have to pull the fan clutch to get to the water pump, and have to unscrew it from the shaft to remove it. (you have to find a way to hold the water pump, which is designed to spin, in order to unscrew the fan.) at 100k plus, would change the water pump if the fan clutch is bad, just because the water pump is likely to go sooner or later.
You don't really have to change them together, but it depends on how much of the job is labor, and how bad you want it fixed for a while.
clicking sound near fan clutch has an interesting discussion. plenty of other discussion on fan clutch here and a few other sites.
you may want to ask another dealer. you may need a software update. there are a few troubleshooting steps you can try, but it sounds like you are not really looking for a DIY project. (you can verify the fan is clicking by disconnecting the connector for the fan clutch, or by pulling a relay)
now if the fan does move (back and forth not in the circle it is supposed to) when you push a bit, the water pump is pretty well gone and you need to at least do that before it dies.