Drinnor
Slytherin, even in canon, the hat wanted to send there, and if he lives with Tonks, then he will not be biased towards this house because of Andromeda. And apparently, Harry’s character suits this house, but Ravenclaw too... Plus Tonks will be able to teach him customs and etiquette as befits the heir of the ancient and noble house of Potter + possibly duels and combat (Andromeda, she is the daughter of the House of Black and knows about it), laws and politics (Andromeda’s husband, if he is a lawyer/lawyer in your story), a partner in duels and help in mastering magic (Dora, or just Tonks as she likes).. . English is not my native language