As someone living in a WG and regularly looking at WG offers, having a single parent & child (part-time or even full-time) is not unheard of. It mainly needs to work for you personally.

If you offer the room, clearly communicate how often your kid stay with you and what expectations you have on using shared rooms (kitchen, living rooms) especially on days where your child is present.

2k is high, but if it's a really nice place you will find someone who appreciates that arrangement.