As anyone who has played either of these games knows, when your main character gets killed, the following happens, in order:
1. The UI disappears to show the main character's death. During this time, the game is still running, though entering orders is not possible.
2. The game shows a video. (You know what video I'm talking about, right?)
3. The prompt to reload appears. At this point, the game is frozen and there is no way to continue without reloading or quitting.

My question is, is there any known method to break this sequence (probably in part 1)?

I can think of two possible ideas:
1. At least in BG2, there exist party transitions that do not require the whole party. (The exit of Irenicus's Dungeon is an example of this). What happens if someone moves there after the main character is killed? (If you get an autosave, what happens when you reload?)
2. What happens if somebody (like a dismissed companion or Limited Wish genie) initiates a conversation at this point?
3. What if somebody steps on a cutscene trigger at this point?

Ideally, I would like a way to continue playing after the main character, or better yet, the entire party, has been killed.

No, there's no way. ** All actions are cleared at the moment of player death. That can't happen. ** That can't happen. ** That can't happen. **

Is it possible for a script to trigger an action after a player's death?

No. Script functions are not executing during player death -- scripts run sequentially.

Actually you can continue playing when the main character dies. It's a bit tricky to pull off, but the one place that comes to my mind is the 3rd level of Naskel mines, where there are traps just by the exit to the next level. Disarm all the traps apart from the last one (closest to the exit) and order the party to exit, ensuring that the main character goes first. He/she will die, but the remaining party members will move to the next area interrupting the death cutscene. You can continue the journey without Charname, but you can't remove him/her from the party, so you can play with just 5 NPCs. I don't know whether it's possible to finish the game that way and it can cause some strange "bugs", because e.g. certain characters will try to speak directly to Charname.

I'm mildly curious about how you expect to do that with no characters. Seems like everything on the map would be in fog of war, though I suppose you could scroll around the map and look at the scenery.