Can a redistricting commission be a legislative house?

The Court hears oral argument this morning in Arizona State Legislature v. Arizona Independent Redistricting Commission, a fascinating case that asks whether a popularly-created redistricting commission is consistent with the requirement that:

The Times, Places and Manner of holding Elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof.

Mike Ramsey has a good roundup of links over at the originalism blog. This seems like a very strong textualist case on its face, though there are enough wrinkles that I haven't fully worked through yet.

Many of the arguments have been about whether "the legislature" can be construed to include the people of the state if they have a general residual power to legislate. But one thing I've been wondering lately is whether the challengers could win by arguing that the redistricting commission should be seen as one house of the state legislature, for constitutional purposes—a third house with a very special and limited set of powers. If so, then maybe the rules are being made by the Arizona legislature because they're being made by the third house. (I assume that the Elections Clause doesn't say how many houses of a multicameral legislature have to be involved.)

I'm not at all sure that argument should work, but it seems more plausible than some of the arguments I have seen, so I thought I'd air it here to see if others have thoughts about why it's wrong.