I don't see the problems you mentioned. Yes, the GW and light travel at c. But they each provide different information.
The GW amplitude "loudness" gives information on the distance to the binary collision, while the light (actually the redshift) provides
information on the velocity. Those two things tell us about the expansion rate at that point in space.
The problem is, this method is not very reliable as the GW that is received may be from different places in the disturbance, throwing off the distance measurement. So they need to do a lot of readings and average them.
Right now, we have two different measurements from the CMB and from the Cepheids and they do not agree. So this will either provide a third reading, and more confusion, or it may confirm one of the other two and settle the question.
I would just average the two readings we have and save a lot of time and money.