A dead battery does complicate shiny hunting in RS. FRLG don’t use an internal battery, so it doesn’t affect them at all. Emerald is weird; I’ll explain that one soon.
In RS, the seed for the RNG is determined by the internal clock. But when the battery runs dry, you get stuck with a fixed seed. Emerald has a glitch where it always has a fixed seed regardless of whether your battery is dead or not. I’ve already talked about what a fixed seed is, how it affects shiny hunting, and how to work around it in this answer, but to sum it up, it doesn’t really have any impact on your shiny chances if you’re hunting via random encounters or hatching eggs, but it does make SRing nearly impossible, so you have to use a workaround for static encounters like legendaries or the starters, which are explained in the link.