Very interesting, speedbug...I would have thought the differences with the A10s version would have gone way deeper than just a u-boot file. Although I don't have any of the A10s devices, I felt motivated to try a little experiment:
(1) I downloads the Fedora 18 image you linked to and followed the instructions to get a working (Hackberry A10) version. Fedora 18 is painfully slow on the Hackberry, but it works.
(2) I took everything in the Fedora first partition and hid it in a new directory (say, FEDORA/)
(3) I copied the contents of the Tinycore armv7 release (today's dCore v5!) first partition into the ex-Fedora first partition, and then
(4) booted to Tinycore.
If you were to do the same for A10s, you might get TinyCore working. Of course, the real solution would be to dig out the u-boot.bin from the Fedora scripts, but I have no incentive to do that.
EDIT: Fedora will run with the TC kernel, so any a10s fixes to the kernel config must be small /EDIT