I'm thinking that beheading/whacking may be my best bet?

I'd say yes, chopping lower and then directional pruning as it grows will give a more tree like silhouette. Obviously one side will have to be cut off to get the most out of what you already have. The left side looks promising.

Lol, just reread what you wrote and looks like the cacti group already suggested the things I was thinking.

OK so, the more mature looking left side, has bar branches, which is typical of jade and many other succulents. For bonsai there is some basic rules that will help you. I like to think of them more as guidelines :). You'll Want to identify your first branch, second branch, back branch for depth, etc.. Trying to create different levels and pads as the tree grows after your main branch decisions.

Some of the easiest things to start with are trimming anything growing directly up or down. For a jade it's going to be probably one leaf of a pair. You'll notice the pairs alternate up the stem, that's where the directional stuff comes in. I'm not sure if I need to elaborate on that, you can let me know?

Maybe just Google, the basic rules of bonsai to get some idea of how it relates to your material. Sometimes pictures are really difficult to give advise for. They are very 2d.

Also, a better draining soil mix would be safer, you won't have to worry about overwatering or rotting the tree. Something with pumice or lava, hard fired clay, small fir or pine bark chips. Even if just adding this to the peaty mix you already have.

The direction of the branches is what you allow: my first step would be to repot into terra cotta with Al's Gritty Mix. When you repot, do so with a future in mind: is there a branch you imagine being a future trunk? Do you want an upright? A multi-trunk? I'm going to bet you find 2 separate plants - do you want to separate them or keep them together? Look at some pics of Jade that are similar to the way your jade is growing to get ideas.

If you want to go in the direction of a bonsai, one rule is that there should not be trifurcations (a single spot where the plant breaks down into 3-branches). I can see at least one spot with trifurcation (A, B & C in pic 1). Remove the branch that is growing toward the center, or too upward/downward. I'd probably remove branch A as it is smaller and the other two give more to work with, but I can only see a pic of the plant - you have the real thing in front of you.

After that, I'd probably just let it grow. You could chop it way back, and get more branching (ramification) lower on the plant, but I'd let it settle into a new pot. There are really two pruning methods: clip n grow, and harder pruning. With the first, just let each branch grow until it has 5-6 sets of leaves (nodes with leaves growing out of each side) and cut/pinch it back to 2-3 branches (do all branches at the same time). With the other approach, you just let the plant grow for a season or two before each prune and trim it back harder, even removing all leaves.