I have found the L298P to be perfectly okay for this application. There have been folks having some heating issues, but I believe that it was due to the weight of their sled vs the weight of the counterbalance. The bungie-cord method does not supply a constant counter-weight to the sled, and therefore the motor currents get quite high. I use water jugs and a couple pulleys to provide about 16 total pounds of counter weight against the 22-pound sled. My motors and drivers barely get warm.
In the source code on the github site you list, there are compile time switches (#define) for the various versions of shield boards (both TLE and L298 based). The Due software requires the EEPROM to store the GRBL parameters and run-time position stuff, and that is why on the schematic above it was added to the Due (I did this using a Mega-breadboard between the Due and the shield.) I did modify the shield (latest at that time) artwork to support the EEPROM and that is what is on the site.
Hope this helps.