Well going back to the origin of the costume, it was first designed as an alternate Batgirl (Babs) costume, so in the very beginning there was no wig and it was just hair.
Then it was repurposed for Kate when the decision was made to 'resurrect' Batwoman, and the design wasn't being used for Babs anyway.
In Kate's early appearances, it was literally her hair, she had long hair back then.
And it wasn't until Rucka with Kate's starring intro (as opposed to her previous guest status) that the hair became a wig. So if reboot-Batwoman wasn't using a wig, and it was just her hair, IMO I don't see the issue.