It's an old post but I like experiments. I heard someone in Hawaii say they didn't have any clay, and all this was said. Maybe selling/ trading it is good experiment. there should be plenty of sand nearish to you to mix to your clay although idk it is about non compacting clay. I'm sure you're not on any volcano but there is maybe some volcanic ash around for roman cement as a natural building material experiment.
or
Try getting some sand and making at least a good sized cob oven or full on kiln of some sort to try firing bricks at higher temperatures. Cob ovens can go up in a few days and get to 725 degrees (farenhiet or celcius idk)maybe more, most cetainly more. there may be some certain temperature that is necessary.
look at this
http://www.ceramicartdaily.net/PMI/KilnFiringChart.pdf
you need over 1000C degrees to start getting melting and fusion into glass and ceramic. 1400C (2550F) for porcelin. Maybe a cob oven with a bellows might get up to something more but i'm sure there are proper kiln designs. campfire can be 900-1100C. You can certainly get that high burning wood, maybe you don't want flame right on the clay and to retain the heat for some time baking and other factors. A straight clay brick will be as breakable as grand daddies porcelin urn over the fireplace or at least the antique vase i'm about to drop. it shouldn't crumble but isn't that strong. Bricks are fired and they're clay, maybe compacting and added with other stuff which is a whole nother process. Mudbricks with sand is still another thing add some volcanic ash or concrete and or some drying agent minimally in the mix, doesn't need to be fired.