In the standard databases you will probably only find the IPR fullerenes of which C60 is the smallest. If you haven't already, check out the calculations by Sergio Díaz-Tendero, https://www.researchgate.net/profile/Sergio-Diaz-Tendero , and coworkers. Since they are probably the biggest experts in non-IPR calculations, you will probably have to read and cite their works, anyway.
In some of their works you may find the Schlegel diagram of the most stable C20 isomer from which you can set up your drawing.