I was thinking Catwoman as the "steady night relationship," but still including the adversarial side at full blast. Maybe show Bruce smiling very briefly when he figures out that he can chase her tonight because it's quiet, have her character embrace the adrenaline junkie aspect while still clearly trying very hard to get away, and have her demonstrate that she is good enough to escape him, but that he knows who she is. Have a personal repartee with a clear undertone of an emotional relationship, and maybe even establish that he occasionally employs her for jobs.
Throw in a scene where they meet at a ball, with the gag being that she has Bruce as her mark and he's using it as an excuse to keep her from picking people's pockets, and throw in a few lines that imply she might realize he's Batman but absolutely refuses to acknowledge that fact because that would seem like "commitment" on her part.
And use her as a control for whatever emotional changes Batman goes through. So if he's actually kind of light hearted at the start of a Killing Joke or Death in the Family storyline, we then get to see the effect it has on him at the end when he's brusque, angry and sad, and she shows her own connection to him by vocally showing him sympathy and maybe returning her ill gotten gains of the night as an apology.
