Looking at photos of her online, she adds highlights to her hair all the time. She might actually have some red highlights in that GCPD photo - but I don't see her going full redhead. Bright red hair is very rare and usually associated with blue eyes and a pale complexion. But there are some reddish brown variations, that don't look like Gordon-red.
There's a full wikipedia article on the topic:
https://en.wikipedia.org/wiki/Red_hair
Some salient points:
"it is associated with fair skin color, lighter eye color, freckles, and sensitivity to ultraviolet light." - that is, almost no ability to tan in the sun, and easily burned.
Red hair is so rare that Barbara Gordon could never have had a successful secret identity with her hair showing. Barbara's red hair on Leslie Grace would be even more rare - one in a million? Probably even more rare than that? Has anyone ever seen that combination? A darker-haired Batgirl in film can more believably keep her identity secret.
I don't think we'll see a Batwoman-type wig. I think we are just going to see a darker-haired Batgirl, maybe with some auburn highlights, and I would prefer a natural look like that to a wig.