Let's just dive in to that juicy backstory, huh? Spoilers for Stormblood ahead.
Law Rose, formerly Laurentus oen Aelius, was a young, rookie member of the XIVth Imperial Legion shortly before the events of Operation Archon. Laurentus was technically spared from the battle at the Praetorium due to still being in training, but believed himself to be at fault for not participating and, at the very least attempting to protect Gaius and his tribunes, whom he saw as role models. The feelings of uselessness led him to seek strength by any means necessary, leading him to volunteer for a series of experiments into the Echo being conducted by Aulus mal Asina. Unlike later, more successful trials, these early attempts were far more, well, experimental. At one point, Laurentus was subject to a surgical procedure on one of his eyes, leaving him with a large scar. With the experiments deemed a failure, Laurentus was sent back to the front lines as a member of the XIIth Imperial Legion. However, shortly after his redeployment, Laurentus disappeared from his post, not to be seen again.
Disillusioned with his country, Laurentus took the persona of Law Rose, wandering Sharlayan mercenary. Law tried, for a time, to maintain neutrality toward world events, but eventually, he would take issue with the sheer amount of Primals wreaking havoc on the world, and dedicate his efforts toward thwarting them every time they should be summoned, and taking preventative measures to halt their summoning in the first place.
As a person, Law is very closed and taciturn. He sees himself as having no home in Eorzea or in Ilsabard. He fears the fallout of being discovered as a Garlean, and as such he is hesitant to make ties with anyone. As a pureblooded Garlean, Law cannot manipulate aether. Because of this, he has grown to see magic as a wild and dangerous practice, more trouble than it's worth. How much of this is simply jealousy, however, is uncertain. During his time with the XIVth, Law was trained in the art of the imperial gunblade, as well as how to pilot and maintain magitek vehicles and other machinery. He also knows his way around a pair of knives, mainly because using an imperial gunblade in open combat may raise suspicion.
Now, the big question I have is: would it make any sense for Law to have some form of artificial echo, or is it strictly canon that Fordola is the first to have sort of success with that? Does it even make sense for Aulus to take on a pureblood Garlean as a guinea pig?
Other than that, though, I think the story should more or less check out? I'll be honest, when I make a character in an existing setting, usually I try to keep them a good distance from the main characters, but it's way too tempting to tie this character in with the XIVth, so I guess I'm a hypocrite.