Now I'm wondering if the part about "enjoyment" wasn't mere psychoanalysis but something I either unusually lack, or which is unusually obscured from my sight.

Enjoyment isn't the right word, I don't think. My wife and I both described the "Bruce effect" sensation as being more like a sense of recognition or rightness -- like confirmation of something that you expected, something that's just the way the world works. That, upon successfully losing, it's like, "yep, this is where I'm supposed to be". Not enjoyment... more like satisfaction... though that's still too strong. Closure, maybe? Relief? It's a brief and subtle reward, not a conscious pleasure.

It was just a loser side with bad habits, probably formed mostly by hyperbolic discounting or poor impulse control.

Anosognosia. Don't speculate, investigate -- observe the automatic thoughts in action, rather than adding voluntary thoughts on top of them.

And it occurred to me that I should give this side a name and separate it out from my real me.

Be careful of how you do that... dispassionate separation is okay, rejection is not. When people actively reject parts of themselves ("that's not me; I would never do that"), they make it more difficult to observe or change the actual motivation involved. ("After all, I would never do that... therefore it's irrational/bad/whatever.")

The way to remove something like this is to imagine what it's like when that part of you gets its wish, so you can get a glimpse of what the reward is. Then you imagine having that reward, and find out what, if any, reward is behind that... and so on, all the way to the root reward emotion, fully experienced in your body, and chain backwards through the same path by which you came, re-experiencing as though you already have the root reward... noticing the difference in available choices, i.e., "If I already have this feeling, do I really need to do X? Is it easier to get X?"

Once you've fully worked back to the starting point, you'll have changed the response options for the original behavior context -- i.e., the original response will no longer be compulsive.

This is a very short sketch of the technique; my shortest training on it takes over an hour and assumes at least a little prior mindhacking experience. (The technique itself can be applied in about 10-20 minutes after some practice, or with the assistance of an instructor/guide.)

There are a few subtleties, not the least of which is that you need to be able to actually pay attention to your autonomous responses without injecting conscious interpretation, speculation, or critique (e.g., "that's stupid, why would I want that?" etc.).