I am imagining these "solar sails" to be essentially spherical, both for structural reasons and so that their response to applied laser power does not depend on orientation. The electronics in precision guided mortars in the US army survive 50,000g.
http://proceedings.ndia.org/C347/Davis.pdf
(This is only for the fraction of a second when they are fired, and I don't know how it generalizes for longer durations.)
I conjecture that the laser profile will be "shaped", if at all possible, so the trajectory of the projectile is somewhat self-correcting: if the projectile starts to get ahead of the planned trajectory, the acceleration will decrease, while if it lags the acceleration will increase. This wouldn't require any realtime feedback by the laser emitter.
With many projectiles and a massive investment in a laser array, there will be some ability to repeatedly attempt this and iterate.
Still, this is extremely aggressive even many decades in the future.
EDIT: Their proposal confirms that the beam will need to be shaped for feedback. ("By modulating the beam modes, get greater uniformity could be attained, and feedback between the nanocrafts and the beamer array would allow real-time adjustments.") No details though.
http://breakthroughinitiatives.org/index.php?controller=Foru...
Rather than the spherical shape a supposed above, they propose a thin sail with some undefined graphene skeleton. No idea how that is supposed to survive 2-minute long explosive accelerations.
http://breakthroughinitiatives.org/index.php?controller=Foru...