No, the one vote per state rule comes from the 12th Amendment and could only be replaced or superseded by a new constitutional Amendment. See the bolded portion of the Amendment below:
... if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President. But in choosing the President, the votes shall be taken by states, the representation from each state having one vote; a quorum for this purpose shall consist of a member or members from two-thirds of the states, and a majority of all the states shall be necessary to a choice ...
This article from the Congressional Research Service provides a bit more detail on how this would work:
Each state, regardless of population, casts a single vote for President in a contingent election. Representatives of states with two or more Representatives would therefore need to conduct an internal poll within their state delegation to decide which candidate would receive the state’s single vote
Contingent Election of the President and Vice President by Congress: Perspectives and Contemporary Analysis
As you point out, the details of how this election would be carried out are based on House rules and can be determined by the party that controls the newly elected House. However the 1 vote per State requirement is laid out in the Constitution and can't be changed.
You mention the issue of "divided" delegations. The way that worked was that a state delegation could only cast a vote a candidate if a majority of the delegation voted for that candidate:
For the first round vote, within state delegations, a majority of state delegation Members present and voting was required to cast the state vote. If a majority was obtained, the name of the preferred candidate was written on the second round ballot. If there was no majority, the second round state ballot was marked “divided.”
A divided ballot could result if the vote was tied, if there were multiple candidates, so no one received a majority, or if enough members of the delegation were absent from the House so that the leading candidate did not receive a majority of the total. While the House could change the rules to allow states to cast a vote for the plurality winner, it seems unlikely that a state would be unable to reach a majority thanks to modern transportation and the strength of the 2 major parties.