I'm late to the party so probably a mute comment.
Maybe I don't understand exactly what your trying to do here but a trench box seems like it will work. You just have dig from inside the box and push it down as you go. The deeper you go the more friction the clean rock exert on the box. You've already mentioned using a ho-pack for compaction it can also be used to drive the box down. Here's the rub. You may find it difficult to get the box out. I'm not sure if the ho-pac will be as efficient when trying to extract. On top of that I'm not sure if a mini can even lift a trench box!
Maybe I don't understand exactly what your trying to do here but a trench box seems like it will work. You just have dig from inside the box and push it down as you go. The deeper you go the more friction the clean rock exert on the box. You've already mentioned using a ho-pack for compaction it can also be used to drive the box down. Here's the rub. You may find it difficult to get the box out. I'm not sure if the ho-pac will be as efficient when trying to extract. On top of that I'm not sure if a mini can even lift a trench box!