I've never heard of a car where you have to remove the instrument cluster to get to the flasher. It *is* tucked up behind the cluster where you are hearing it from, but it is normally accessible from underneath.
It's a pain in the ass (literally) but you have to open the door, pad the door sill with a blanket, and lay on your back in the doorway to see what you're doing. If you plan to be under there a while and need support for your legs, go get a few milk crates or something about the height of the door jamb. Turn on the flashers and you can locate the little clicking black box fast.
What you might have to remove is not the cluster, some cars have a trim panel where your knees are that keeps the wires and components from getting accidentally broken if the driver started kicking.
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