Do the Constitution and applicable federal laws allow states to elect representatives in this manner?
The text of the Constitution is silent on the issue, though Justices on the Supreme Court have disagreed on that point (see below). However, ArtI.S2.C1.2, Electors for the House, presents a possible Constitutional problem with the proposed scheme, in that, "Members chosen ... by the People" is qualitatively different than electing a slate of Representatives chosen by a party (or party primary). In particular, independent voters would be excluded from the choice of representation.
The House of Representatives shall be composed of Members chosen every second Year by the People of the several States, and the Electors in each State shall have the Qualifications requisite for Electors of the most numerous Branch of the State Legislature.
With respect to the law, 2 U.S. Code § 2c., Number of Congressional Districts; number of Representatives from each District; declares that "no district [is] to elect more than one Representative".
Historically, Representatives were elected "at-large". However, in 1842, Congress passed a law requiring "single-member districting". In 1872, the law was amended to require "equally populated districts".
See, ArtI.S2.C1.1 Organization of the House of Representatives
The Court's disagreement
For the Court in Wesberry, Justice Black argued that a reading of the debates of the Constitutional Convention conclusively demonstrated that the Framers had meant, in using the phrase "by the People," to guarantee equality of representation in the election of Members of the House of Representatives. Justice Harlan in dissent argued that the statements on which the majority relied had uniformly been in the context of the Great Compromise – Senate representation of the states with Members elected by the state legislatures, House representation according to the population of the states, qualified by the guarantee of at least one Member per state and the counting of slaves as three-fifths of persons – and not at all in the context of intrastate districting. Further, he thought the Convention debates clear to the effect that Article I, § 4, had vested exclusive control over state districting practices in Congress, and that the Court action overrode a congressional decision not to require equally populated districts.