First I'll say I'm not a laser guy but am a Grbl user on milling machine and am active on the Grbl github site. I have been thinking about trying laser out on my mill. anyway, here is what I know:
Grbl in general is stable and rock solid and has 2 current repositories that are kept up to date. One is for the Uno (328p) type platforms and the other is for the Mega 2560 platform. These 2 repositories are kept up to date.
There is a large laser user base and many of the things implemented are laser specific. I occasionally see posts about laser topics and most are configuration questions, not problems with how laser mode works. As I understand it, the laser mode has many features and can be read about here: https://github.com/gnea/grbl/wiki/Grbl-v1.1-Laser-Mode
As for using on the DUE, I'm sure there are ports, but who knows how up to date they are, or if maintained at all. The main programmer for Grbl is working on an ARM port with a HAL so that porting to any ARM platform should be much easier. For now the "official" versions are 328p and mega 2560. That said, the Mega2560 may work for you and only costs a few dollars. It is compatible with the RAMPS boards already.
Here is the Mega2560 verion on github https://github.com/gnea/grbl-Mega/wiki