At the end of a catching show, the game breaks down the final score into three factors before telling you the overall score.
This factor is not dependent on any aspect of your performance, only on the Pokemon you brought in. Each species has a base score value, which falls between 30 (Caterpie, Pidgey, Magikarp, etc.) and 100 (Mew, Celebi, Jirachi, Deoxys). This score value is inversely proportional to how often it can be encountered. The Catching Points that are displayed is simply the sum of the scores for all six migrated Pokemon, and is therefore constrained to be between 180 and 600.
This is the most performance-based (and luck-based) factor. From the moment the "Start" message appears after the countdown, time is ticking, as are Time Points. The invisible gauge starts at 2000 points, and with every passing second, you lose 2 points. Once the timer reaches 16:40, the gauge runs out of points, and stays at 0 forever after instead of becoming negative. So the lowest score possible is 0, and as for the highest score, an immediate encounter and ball throw takes about 27 seconds before you can start moving again. Add in time to deal with the text box at the end, and you could probably get through a show in about 2:45, yielding Time Points of 1670.
This factor can be broken down into two other sub-factors, neither of which is displayed individually by the game. One of them, like Capture Points, is set in stone the moment you select a set of six Pokemon to migrate, and nothing about the way the Capture Show plays out can possibly cause it to change: for each different type that's represented among the six Pokemon as a whole, you get 50 points (so at maximum, you could have six different dual-types to cover 12 types, good for 600 points). Another way of thinking about this is that dual-type Pokemon inherently have an extra 50 points on top of their capture score, that mono-types can never match. As for the other sub-factor, when you catch a Pokemon in the course of the show (other than the first), if it doesn't share any types with the Pokemon you caught immediately before it, you get a bonus of 200 points. So, for example, if you caught four Pokemon in the order Bulbasaur-Venusaur-Charmander-Charizard, that would only count as one 200-point bonus (Charmander doesn't share any types with Venusaur), but if you caught them in the order Bulbasaur-Charmander-Venusaur-Charizard instead, then you've split apart the duplicate typings, and get three 200-point bonuses instead. The lowest possible score in Type Points is 50 (if you migrate six mono-typed Pokemon that all have the same type), and the highest is 1600 (if you hit those 12 different types, and therefore are guaranteed to receive five 200-point bonuses no matter what order you encounter them in).
As far as overall score, the individual components at their maximums add up to 4200, but it's plain to see that this upper bound is impossible. The only way to get a Capture Score of 600 is by transferring six mythicals, and the only mythicals that existed as of the GBA era all had the Psychic type in common, so you wouldn't be able to get even a single 200-point bonus with that squad. On top of that, the obvious fact that encounters take time means the 2000 starting time bonus is impossible to hold onto.
In order to get the highest possible score, you'll need a hefty dose of good luck in any case, but it appears that the single best roster of Pokemon you can migrate is Scizor (Field), Blaziken (Field), Rayquaza (Field), Gengar (Forest), Celebi (Forest), and Tyranitar (Mountain). This accounts for a sum of 2150 between Capture Points (550) and Type Points (1600), which is the highest possible sum you can get among those two components; further, it avoids having to visit the water areas at all (because the dialog boxes and use of Surf are slower than running through grass) and lets you enter the Mountain area with only one Pokemon left to catch, and ideally encounter it on the first step into the grass in that area, minimizing travel time. The odds of passing each check to get an encounter at all with these high-scoring (and thus rarely-encountered) Pokemon are 9/29, then 3/13, then 3/23 in the field, either 1/8 followed by 1/21 or 1/24 followed by 3/23 (depending on the order) in the forest, and 3/23 in the mountain, but assuming you got perfect luck to encounter all of them without having to wait any longer than it naturally takes to run the shortest possible path through those areas, your total score would likely be in the low 3800s.