Instead you let a minority control the majority
Myth. The House vetos the minority from controlling the majority. That's the purpose of the House.
Veto powers should be limited to that which directly impacts each state materially, not simply everything including things happening afar
If the majority wants to enact something without the consent of the Senate, they are free to enact it at the state level: For themselves, rather than imposed against a minority populace that would reject it for themselves.
If that is not a viable option, then the federal government has too much power and needs to be reined in.