Devices like the SH-101 were designed to work with external controlled voltage for synchronization with external devices, which is called "CV Gate". It is one method that people used before MIDI and the MIDI-clock specification (or SMPTE time code, for that matter).
I do not believe there are any metronomes, per se, that provide CV Gate, but back in the day there were many devices on the market that could be used for precisely synchronizing analog synthesizers and analog sequencers with each other. There are more recently-manufactured devices that can provide an analog synchronization that also can be interfaced with a MIDI signal, so you could get your analog SH-101 in sync with a DAW like ProTools or Logic or Cubase.
I did a quick Google search and found a reference to several devices for providing an interface between CV and MIDI systems that are currently manufactured and sold by the Kenton company in the United Kingdom. Here is a link.
I have read that there is a company that sells a retrofit kit for the SH-101 that gives it MIDI input and output capability, which would probably permit syncing over MIDI. That would be a good solution if it is affordable.
After a quick Google search, I came across references to using an audio output from a Roland drum machine of the same vintage (say a snare drum sound from a TR-707) and running that into the "Click In" input you described on the SH-101, with the appropriate analog audio cable. I assume that a Roland drum machine would enable you to choose a precise, exact tempo. But if you want to synchronize with other systems, a more modern CV-to-MIDI interface would be the way to go.