Which kernel does HORIZONS get the Geometric Albedo from?
As barrycarter commented, codes_300ast_20100725
has ephemerides for 300 large asteroids, generated in 2010 and supposedly valid from 1800 to 2199.
For other asteroids, a Celestia developer forum shows how to how to request an SPK from HORIZONS's telnet interface.
However, they appear to use the SPK for ephemeris only, adding albedo in Celestia's own SSC file format.
In SPICE documentation, neither the tutorials nor the "required reading" seems to mention albedo.

A more direct approach to your goal might be the JPL Small Body Database Search Engine.
For example, if you query for numbered NEO asteroids with absolute magnitude H &lt; 16, you get a table of 169 rows.
Diameter and albedo are known for some asteroids but not others, even though their orbits and rotational periods may be well established.