I don't have a tested idea, but I think the Christmas-Tree idea is a good start. Since you will have things outside, I wouldn't recycle an existing tree for strength purposes. Use a thicker piece of wood as the core. For storage, you might just wrap up the tree with rope, to squeeze the branches in. You might be able to stuff several into a zip-up type of Christmas trees storage bag. If you put an L-shape bend at the base of a branch, you might hinge it to the tree center so that it will bend up, but not bend down. However, that hinge would have to be near the surface of the foam to permit that. So, you might have to add wood spacers to the trunk. I will attach a quick sketch of what I am thinking. The hinge is a pair of "U Nails". I suggest getting the barbed kind)
I wouldn't use any hard coating on the outside. Perhaps use pool-noodles to build thickness. You can cut the noodles into strips and wind them around your core. Hot glue works to some extent, but only if you use a glue gun that can set for low-temp. To cover, I would use dark brown fabric and attach it with something like 3M's 77 or 90 spray contact cement. I've even seen bark print on a fabric in the past. The example you gave has things twisting around, and I think that is a great approach. It will really help things hold together.
I am not sure what convenient hardware-store wire to recommend for the core of the branches. I suspect that drop-ceiling wire is too thin. And for the spacers. I drew square blocks of wood. But, I suspect that they should be rounded to form the knob that often forms where a tree branch attaches. Your own miniature has a pretty substantial root spread at the base. But, I suspect that you can just omit that, and save a lot of storage space. Also, I would make the tree trunk and branches thinner for the sake of storage.
View attachment 775112