I just had a bit of a eureka moment regarding how to structure the opening scene. According to my current plan, you only need to make one major change in order for the opening to work: make it seem like the Trade Feds know what’s going on from the very beginning. They’re immediately aware that the ambassadors are Jedi, and before Obi-Wan and Qui-Gon have even entered the conference room they’ve already made plans for what to do next.

Here’s the way I want to structure the opening:

Cut straight from the opening shot to the captain saying “With all due respect…” removing redundant dialogue.
After the Jedi ship lands in the hangar, cut to the first appearance of Sidious. This makes the Trade Feds seem more proactive, establishes that there’s a mysterious force at play here from the very beginning, and increases the tension in the conference scenes (since we know that the Trade Feds are planning to kill the Jedi). Also, cut the bit about making the invasion legal, as part of a general change to make the Trade Feds seem more like bullies: they know what they’re doing is illegal, but they think the Republic is too weak to stop them.
Then, cut to the conference door opening, with TC-14 saying “We are greatly honored by your visit.” Obi-Wan and Qui-Gon’s conversation plays out in full, with Qui-Gon’s lines now having an added element of dramatic irony.
Afterwards, cut to the first appearance of the Queen, as she has a tense conversation with the Trade Feds. Gunray’s denial that there are any ambassadors onboard now has a greater ominousness to it, since they still think their murder of the Jedi will succeed. Cut Gunray’s line “We would never do anything without the approval of the Senate”, as part of giving the Trade Feds a “screw you, you can’t stop us” attitude.
Continue the scene the way it did originally, with Gunray cutting off all communications, resulting in Palpatine’s hologram shorting out. However, end the scene after the governor says “Invasion.” This both ends the scene in a much stronger way, and removes more talk about how the Trade Feds always adhere to galactic law.
Next, cut to the Jedi ship being blown up, and toxic gas flooding the conference room. The action scene on board the command ship plays out mostly in full, but with the standard cuts (no “Destroy what’s left of them”, less Trade Fed cowardice).
When Obi-Wan and Qui-Gon see the invasion army, cut Qui-Gon saying “This is an odd play for the Trade Federation.” Again, the Trade Feds aren’t concerned about legality. After Obi-Wan’s one-liner about how “The negotiations were short”, cut to the start of the invasion, and proceed from there.

So, does that work? It’s not a radical change like I had previously envisioned, but I think it really helps in terms of making the opening more engaging.