I think it makes sense as a mechanic and doesn't darken the tone of the game at all. Adding the cops to the game gives you more obstacles to overcome, which keeps things interesting, but also wouldn't darken things so long as you control what exactly happens after she's caught. Like for example, if you show our protagonist being released from police custody a couple hours/days later with a slap on the wrist, that wouldn't be very dark or life-ruining at all. Unrealistic, sure, but it's a video game and realism isn't that big a deal. If them being let off easy is too unrealistic for you, just say that they escaped police custody somehow and get right back to streaking. All that really matters is how you portray the whole situation, not necessarily how it would play out in real life.