xf86-video-ati is the correct driver, but it is unloaded for some reason - perhaps you need the graphics-4.19.10-tinycore64 extension to be loaded or, if that doesn't help, Xorg-7.7-3d.
Note that using the amd drivers and firmware is not going to work and using the TinyCorePure64 iso instead of the CorePure64 iso does not help either.
Try CorePure64-10.1 + graphics-4.19.10-tinycore64 + Xorg-7.7 + xf86-video-ati + flwm + wbar + aterm