If they really went all out with it, really went there and pushed it to be something new and different from the main series, she could easily be like Absolute or Ultimate Doctor Who, still recognisably Doctor Who but different enough to go places the show can't
Maybe what the New Wilderness Era needs is a series of dark, gritty, more grown up novels featuring the Fugitive Doctor that tell stories of her missions for The Division. No monsters and villains from the 1st Doctor onwards, give us new ideas!