After doing the quests with Melly Teletone and Pedgi , you can talk to Melly upstairs and get [Melly's Metronographer+++] item. When you use it, a blob will appear next to Bob the innkeeper.
The blob is a vendor who sells Infinitea item. When you drink you have 30 sec buff.
Go to 54.64 85.86 under the effect of the tea and click the clock Pendulum clock by the waterfall and the rare will appear.
The rare has 3000k~ hp. After the timer is over the buff the rare still stays but the clock is gone.
The rare is daily as you can click on the clock only once per day.
So far the rare dropped only trash items, but I suspect it might drop [Investi-gator's Pocketwatch] or there is another thing needed to do to make it drop.