No, it doesn't work like that. The military is of course tight-lipped about exactly how powerful SHiELD is, but at least one site claims that the goal is 50kw [1]. That's a lot of power. A few years ago, a 30kw laser test made the rounds in news outlets for destroying a few targets during testing. [2]
Mirrors don't perfectly reflect light in all frequencies. Mirrors are designed to reflect mostly visible light, lasers at this power level are infrared. Any imperfections in the reflective surface would immediately heat up, damaging the reflective surface, which would then heat up even more. And even if it were perfect, and were designed to reflect whatever frequency of radiation the laser's putting out, it's still not going to reflect 100% of the laser's output, and you've probably got yourself a material that's pretty impractical for a rocket.
Mirrors don't perfectly reflect light in all frequencies. Mirrors are designed to reflect mostly visible light, lasers at this power level are infrared. Any imperfections in the reflective surface would immediately heat up, damaging the reflective surface, which would then heat up even more. And even if it were perfect, and were designed to reflect whatever frequency of radiation the laser's putting out, it's still not going to reflect 100% of the laser's output, and you've probably got yourself a material that's pretty impractical for a rocket.
[1]: https://phys.org/news/2018-03-air-fighter-mounted-laser-summ...
[2]: https://newatlas.com/lockheed-martin-laser-truck/36377/