Before you go ahead with any invasive surgery here, have you checked the tension on your TV cable? When you lift the air cleaner off the air horn of the carburetor or TBI, you should see a bracket that supports two cables. The upper one is the accelerator cable and the lower is the TV cable. All this has been covered in earlier threads and on You-B00b but to summarize, the Th200R4 does not use vacuum to tell it when to shift, it relies on internal pressure, which is why the video suggests to swap out that switch. When you attempt to accelerate, the movement of the throttle cable also causes the TV cable to react. If the cable is too tight or slack, the shift won't come at the correct time. Too tight and it shifts too early and very hard; too slack and the shifts are late or not at all.
if you do find that cable is mis-tensioned, then it can be adjusted, and I did post a how to for this last year at some point so the local search engine ought to be able to retrieve it for you, or it can be found in one of the service manuals that Haynes or Chilton publish. If there is no obvious slack or extreme tension then the cable can still be at fault if it is damaged inside its outer cover or has gotten rusty. The cable can be swapped out for a new one without removing the t-mission but the cute little metric bolt that holds the cable boss to the t- case is a grade two item and very soft and prone to be eroded due to age and road exposure.
The reason that I suggest this as a to do first is that, as the video implies, accessing the internals of the t-mission is an invasive exercise that the commentator is doing while the car is on a lift; in short, under ideal conditions. While this could be done in a driveway or in a one car garage, if anything goes wrong or you break something else, or the oil that comes out of the pan is black and smelly and there is a lot of debris in the pan bottom, then you have to figure out how to get the car to a shop that deals with auto-stick boxes and have them take over the job. That could cost you as most shops like to do the whole thing from step one and not get called in halfway through because something went wrong.
Not trying to discourage you here; good to see you want to take a run at it and wrench your own ride. Just be aware that videos can skip things under the assumption that the viewer already has some experience or has access to reference material for assistance and that they do tend to get made, mostly, under fairly ideal conditions. You don't often see a video of someone in a pick a part yard slithering under a hulk to grab parts because dealing with rust and corroded parts and nasty working conditions aren't things a lot of people want to see. In addition, when you do come across them, they are not live action, meaning that the video was likely edited for both time and the removal of minor details like prolonged or intense bouts of cursing and swearing when something, a bolt or nut, refuses to move or breaks, or a part or tool gets lost, or when the inevitable damage to a finger or hand occurs. (Not planning on mentioning the possible need to deal with resident varmints of various kinds and levels of danger here)
Pictures and progress posts are always enjoyed if you get the time to post them.
Good luck,
Nick