I don’t want to discuss here the wisdom of such a move, only its legality. Could a state simply have each party put up a slate of candidates, chosen by primary or otherwise and the party that wins the state simply elects its entire slate? We seem to be tending that way anyway.
I just read the relevant section of the Constitution, which is article 1 section 2.
https://constitution.congress.gov/browse/article-1/section-2/
Based on my reading, it certainly seems as if choosing members for the House of Representatives in the way you describe would not be unconstitutional. AFAICT the only requirement would be that the state in question would have to pass a law to that effect.
Or, as an alternative, if the Ds get 60% of the vote and the Rs get 40%, that’s the split of the states delegation. (To the nearest whole seat and making appropriate allowances for third, or even fourth, parties.)
It seems to me that I recall a court case striking down proportional representation for congressmembers, but I don’t remember the name. If so, then presumably the same arguments would also apply against winner-takes-all.