Just a few hours ago the moon-bound spacecraft Chandrayaan-3 was launched.
While it will orbit Earth for a few days, it is accelerating/changing its orbit often to prepare for the transfer to a lunar orbit, so its coordinates (especially from a specific location on Earth's surface) are very unpredictable.
Neither an orbital element approach nor TLEs will work here reliably unless they are updated very frequently (the latter only seem to work for earth-orbiting satellites anyway).
In the future, there will probably exist more moon-orbiting spacecraft like the Lunar Gateway which will also change its orbit often. Does a publicly accessible website/API exist that provides the current coordinates for such objects?
In this case it seems ESA's Estrack network is supporting the tracking of this probe, but it doesn't seem like they have a public API: https://www.esa.int/Enabling_Support/Operations/ESA_ground_stations_support_Chandrayaan-3_Moon_mission
Edit: Permalink to the code I adapted from PM 2Ring below, to color observed orbit segments