In this sentence, they are talking about buying a physical location (a house), they are not calling it their home.
Why is うち used?
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This may not be the best explanation but here is my crack on it.
いえ means house/home but more emphasis just on being a location and more physical aspect so I tend to process it’s meaning as house and no emphasis on like emotional ties.
うち means house/home but more emphasis on your house/home so while its a physical location there is more of an emotional tie to it so I tend to process it’s meaning more towards home.
Now this sentence I feel like could be translated into “I am buying this house!” and therefore use いえ but with how it is written with the other words and seems slightly emotional so うち seems a bit more natural?
This one is hard because either way it would not be wrong but there seems to be a slight nuance and deeper meaning which can be hard to pickup when you do not know other words in the sentence for better context. I struggle with this on many other vocab too but hopefully this helps a little!