I'm not saying he should downsize, in fact I'm not making any value judgement about your brother's financial situation. What I am saying is consider this from your niece's perspective. Dealing with this through mediation offers the best chance to navigate the situation with the best outcome for the child.
I get that he's your brother and I am sure you want to fight his corner. I'm just pointing out there are wiser ways to do this. Also try to fight your neice's corner too. Just the title of this thread referring to his ex as greedy is poisoning the well.
I'm not saying you have to like her, or condone her cheating but whatever else she is she is your niece's mother. As such it's crucial she grows up in an environment where there is no toxicity directed at either parent. She will build her initial identity based on both her parents. Seeing them in constant conflict will create a schism in her own psyche.
Like I said I'm in a similar position, my ex will let me have my child all day during the week frequently, but is extremely reluctant to surrender many nights, because she needs the money, and although we are neither one's favourite person I wouldn't characterise it as greedy. It is if anything a survival thing.
I recognise it's not fair that the situation defaults to your brother's ex's choice how many nights he gets which in turn affects the finances, but I would counsel against court and work within the confines of the status quo. I would put a pin in what is or isn't fair to the parents and put the daughter front and centre.
That may mean a reduction in payments to the ex so your brother can fund a place big enough, but when you frame the central premise of your thread "greedy ex" it betrays a non-child centric motivation imho.
I'm just advocating going back to the drawing board and having a look at everyone's motivation. Including yours. By all means listen to your brother's frustrations and sympathise, but try and steer your advise and questions to what is best for his daughter. They will all thank you for it in the end!