The reason why Chris (Sacred Heart) take the form of a rabbit in Standby Mode is most likely because the rabbit form is based off from a present Vivio got when she was young, something very precious to her.
In Episode 13 of StrikerS when Vivio is being treated after being rescued by the Riot Force 6 Forwards Nanoha buys a white rabbit she saw in the hall and returns to Vivio's room placing it by her pillow.
Later in the episode Vivio is seen crying when Nanoha goes to leave to learn of the reason why Riot Force 6 was created. when Fate approaches Vivio to calm her down she picks up Vivio's rabbit, while she talks you'll notice that as Fate moves the rabbit around Vivio's follows it with her head.
you'll also notice that the 2 look very similar (albeit the glow on the cheeks)
Left: Vivio in Strikers holding her rabbit toy.
Right: Magical Girl Lyrical Nanoha ViVid Poster with Vivio and Chris
i would suspect the rabbit has the same kind of sentimental value as Nanoha's and Fate's hair ribbons they traded to one another at the very end of the first season, but later on for a grown woman to be carrying a rabbit toy would seem strange1 so making it her device's Standby Mode preserves it's spot in Vivio's life and thus keeping the memory of Nanoha close to her no matter how far they be apart
1: Vita carrying the rabbit Hayate brought her isn't strange because Vita will still look like a child