When I lived in New Jersey, I used bagasse as bedding for my dogs. After it was soiled, I tilled it into the garden beds where it decomposed fairly quickly. Bagasse is chopped up sugar cane that is left after the juice has been extracted, so it's the whole cane, not just the tough outer part. It was softer and more absorbent than wood chips.
Now living in Hawaii, I squeeze my own homegrown cane for juice, so I know exactly what you're talking about. It's the tough outer rind of the cane. Real tough. It takes a lot longer than bagasse to decompose. Rather than toss it into my
compost pile, I will dig a trench along side a garden row and line it with the cane stalk. Then cover it over with dirt. It will eventually decompose there, but it takes a few months. Like you, I find it is a pain to try to dig the soil where I buried the cane stalks, thus the reason I choose to bury it in a walkway or beside the garden rows rather than inside the beds.
Since you are working with degraded soil due to deforestation, you will need to dig plenty of organic material into your clay and keep the surface covered from the tropical sunlight. I should think that the cane strips could be used as a top mulch which could be pulled aside when you work a garden bed, then pulled back into place for the mulching effect.
If you happen to plant any bananas or fruit trees, I think that the cane strips would be a good top mulch around the trees.