US "parties" are giant coalitions compared to the "parties" in parliamentary democracies. You're solving a problem that doesn't exist.

Change the American voting system tomorrow and legislators will belong to different nominal parties that end up forming precisely the same coalitions.

Love him or hate him, Trump is a great example of this - in 2016, Trump effectively formed a new party focused on anti-immigration and protectionism, which rapidly grew to dominate the "conservative" coalition. But those other parties, ranging from libertarians to the Chamber of Commerce (highly pro immigration and highly pro free trade) parties are still there in the coalition.