Temporally Flat One-Way Quantum Computing
It doesn't make sense to me to ask about non-commuting operators being implemented in a single time step (though constant depth certainly makes sense). However, you can implement non-commuting gates on the logical subspace of an MBQC which are implemented using commuting measurements on the resource state, though the implemented gates are not deterministic.

Actually, I believe you are viewing IQP more narrowly than you probably should. The answer to your question is that any MBQC which can be implemented in a single measurement layer in MBQC is contained in IQP. This is simply because rather than expressing the result in terms of the logical Hilbert space, you can express it as a series of commuting operations on the physical qubits. Shepherd and Bremner actually deal with this in their paper (in section 5.2 where such operations are called graph-programs).