Part of the confusion here is that NALLS is *both* an IAF and FAF.

Irrelevant. The fix definition doesn't change by how you're using it.

You can't use the LOC in lieu of DME for this purpose, due to the
possibility of encountering a false lobe.

This was speculation. Nothing on the approach says that this is true.
I spoke with someone with Flight Procedures today about another
approach, and mentioned the possibility of "false lobes" on a
transition. He indicated that a transition is flight checked for
false lobes and if found, would not be approved. I'm sure there are
exceptions. He mentioned some procedures at Grand Junction that
contain lots of warnings about the possibility.