Bobson cleanly laid out what the constitutional requirements on the general election side of things are: namely that it is essentially in Congress' hands assuming that Representatives are still apportioned by states in a manner as allowed by the constitution.
The other question is the constitutionality of Representatives having fractional votes. While it is clearly stated in Article 1, Section 3, Clause 1 of the Constitution that
[...] each Senator shall have one Vote.
I was not able to find a corresponding clause or implication applying to Representatives. At best, there are the provisions of Article 1, Section 5 which hint at it:
Each House shall be the Judge of the Elections, Returns and Qualifications of its own Members, and a Majority of each shall constitute a Quorum to do Business; but a smaller Number may adjourn from day to day, and may be authorized to compel the Attendance of absent Members, in such Manner, and under such Penalties as each House may provide.
Each House may determine the Rules of its Proceedings, punish its Members for disorderly Behaviour, and, with the Concurrence of two thirds, expel a Member.
Each House shall keep a Journal of its Proceedings, and from time to time publish the same, excepting such Parts as may in their Judgment require Secrecy; and the Yeas and Nays of the Members of either House on any question shall, at the Desire of one fifth of those Present, be entered on the Journal.
Neither House, during the Session of Congress, shall, without the Consent of the other, adjourn for more than three days, nor to any other Place than that in which the two Houses shall be sitting.
Clause 1 states that 'a Majority [...] shall constitute a Quorum to do Business' and that 'a smaller Number' has additional powers. Clause 2 speaks of the 'Concurrence of two thirds'. Clause 3 suggests that 'the Yeas and Nays of the Members of either House on any question shall, at the Desire of one fifth of those Present, be entered on the Journal' – the second half seems rather clear at counting number of Representatives but the former does not necessarily require their votes to be counted equally.
As I am not a lawyer – even less a constitutional lawyer – I cannot provide a discussion in sufficient depth to conclude whether these provisions allow or disallow Representatives having fractional votes in the House. I can see compelling legal arguments on either side of the issue.
Congress (i.e. mostly the House), obviously, would have the power to pass appropriate legislation or orders of business. However, I believe their constitutionality would be challenged and it would likely be up to the Supreme Court to decide whether it would ultimately be constitutional.