Slowly turn a planisphere at the sidereal rate
At the risk of oversimplifying the problem, here is a suggested sequence of steps that may serve the purpose, perhaps with some empirically determined tweaks:


Buy an inexpensive electronic wall clock movement, such as those sold for DIY clocks - look for one using older technology, i.e. discrete crystal and possibly (though unlikely) through-hole parts, plus an electromagnetically driven rotation actuator.
By replacing the crystal on the board with ones of higher and lower frequency, determine what range of possible tick rates the clock movement can work with.
Calculate convenient gear reduction ratios for a tick rate within this range, as required for a gear or pair of gears to reduce the hours (or minutes) actual rotation rate to the desired one

At the simplest, a 1 minute tick rate (no crystal changes) and a 1:10 gear reduction should provide the required rotation.

Get your friendly neighborhood laser cutting shop to cut you the required gears, in a fairly large size (since precision is not essential, and bigger gear teeth can tolerate laser cutting imprecision better).
Attach these gears to (a) the hours or minutes shaft, and (b) a shaft on the planisphere disc, to mesh without too much strain.


While this isn't actually an electronic design solution, hence perhaps not ideal for this site, it addresses the problem through minimal engineering and readily accessible tools.