The video don't look like its really showing much of my problems. Except a cut out in the square 8.
That definitely sounds like fuel starvation, not a "Fox burp" type quit. If you have only a short time to work on it, I would suggest a Sullivan RST-4 clunk tank, set up for suction. It's inexpensive, available everywhere, and should definely allow the engine to run through the maneuvers. You will have to be a bit more careful about fuel measurements, because you can't cut it off by your actions. And recall that merely getting through the entire flight without crashing/underrunning/overrunning/losing pattern points will usually put you in the mix to win in Intermediate because many of your competitors will screw it up somehow.
Just a coaching note, nothing to do with the engine run - watch yourself on getting anxious or impatient in the repetitive maneuvers, because you are consistently allowing the sizes to get smaller as the maneuvers go on. This makes it much more difficult, because the airplane is slowing down even at the best of times, and you are squeezing it tighter, so it slows down even faster.
That may be an element of the engine issue, too, because your are really jamming it and moving very slowly at the end of the square 8. I would speculate that the greatly reduced speed, the altitude, and the extreme drag of maneuver might be causing the fuel to slosh forward enough to uncover the pickup. Not to beat the same dead horse again, but - with the gigantic flaps acting like airbrakes, and the wildly non-linear control response, that exacerbates the effect. But you definitely should expect the engine to keep running.
Your maneuver appear to be generally small, which is good in and of itself, but if the airplane, or your skills, aren't up to the task of dealing with the effects, you might be better off making them bigger until you get a bit more sorted out, and for sure, don't let them get smaller as they go along.
You are doing pretty good, I think that looks like a competitive intermediate flight, as long as you can keep the engine running the whole time and not forget anything.
Brett