Honestly, the battle droids and general evilness alone should be enough to connect the Separatists and the Trade Federation. Both organizations are hilariously generic in the sense that we learn practically nothing about their precise motivations. All that is given in the movies is that the Trade Federation doesn’t want to pay taxes because they are greedy, and the Separatists? Well, they are characterized by their dislike for the Republic and the Jedi. Based on the abundance of big companies present at Dooku’s meeting, we can infer that they share the same bland motivation as the Trade Federation, money.
So, I think the distinction between the two can be put in the crawl of ROTS, without losing anything.
Actually, the bigger question is why Dooku is on their team. His supposed motivation to leave the Jedi and the Republic is the corruption of the senate, but the first thing he does after leaving is founding the League of the Uber-corrupt, super-capitalist of the galaxy. It just doesn’t make any sense. I mean, what greater good could justify this absurd action against his actual beliefs?
Yeah, visually it’s a no brainer. All the battle droids, droidekas, and neimodians are there from the start of ROTS. I just mean if I’m putting the episode 2 stuff at the end of TPM. When Palpatine says that he’s making an army to counter the increasing threats of the Separatists, it should be established that the Trade Federation is one of the groups of Separatists he’s talking about. They’re part of a trend. If anything I think it would make sense to put it in the crawl of TPM that the Federation is a Separatist group, and then reference it at least once or twice somewhere else.
For Dooku I won’t get into any detail. He’s just Sidious’s apprentice next in line after Maul. It’s a shame to leave out all the subtlety but as you said it’s not well put together anyway. His identity is tied too much to the AotC bonkers mystery plot and I’m trying to cut out 98% of that movie.