The Kawamoto's won't let Rei buy it.
Hell, they didn't even let him pay for his own food when visiting despite their limited finance. Even considering he's Hina's boyfriend now, I doubt Akari would let him buy an entire house.
At best, they could accept his help in case things go the wrong way and they lose the house as collateral.
With that said, give me my Kiriyama vs Nikaidou game soon pls thx.