Just read through part 1. I really like the idea of FUN values being multiple timelines, and the connection between the two sets of blueprints we see.

That said, I'm going to be being nitpicking and probably a bit too critical in this post. I do think it's a good theory overall, just giving my feedback on some of it.

Not only that, but their accidents all stem from fear of something.

[...] afraid that Undyne might not be as cool as they thought

I think this is stretching it quite far just to try to make it work. It's hard to find a significant character that isn't afraid of something or other, so I wouldn't really say it connects them.

Your fun value indicates which timeline you’re in. It’s set randomly at the beginning of the game, and may change when you’ve gotten to a “point of no return” on genocide and pacifist runs- Points where you can no longer exit a a certain run, or timeline.

Not quite sure I get this. It changes after you're locked into a run? Shouldn't it be changing up to the point at which it can no longer change if it indicates what timeline you're in?

and they’re all (with the exception of Goner Kid, who’s an anomaly all on their own) [...]

Spotting connections between four characters, then disregarding one of them as an "anomaly" because they don't fit, isn't all too convincing.

And when you also consider both the name of Sans’ “Gaster Blasters” and their shape as compared to the shape of the DTE machine…

I don't really see any connection between the DTE machine and Gaster Blasters to be honest. The head shape is different, the eyes are different, the mouth is different, nothing really looks the same at all other than it being a symmetrical head. I'm pretty sure that it's meant to look like Omega Flowey rather than Gaster.

[...] falling into a powerplant doesn’t fit very well.

Being a scientist doesn't mean that you have good balance. I personally think the core fits pretty well as his death place, being his own creation and so mysterious/magical.

What if, upon creating the DTE machine and discovering the substance that is “determination” within a human soul

Where would he get the human soul from? Chara was the first human, their soul was taken by Asriel, Alphys was scientist around the time Asriel died, and Gaster was dead long before Alphys became royal scientist.

So he builds his greatest invention yet- what essentially boils down to a time machine.

But something goes wrong- perhaps he found what “determination” does to monsters, maybe he saw something he didn’t want to see in the future timelines. Maybe he was even listening into the future at the wrong time..

..And he gets scared. He isn’t careful enough. He makes a mistake.

And suddenly, the world is falling apart. The timeline is destabilizing, nothing has gone according to plan-

Until suddenly, it all ends. The timeline collapses, and shatters. The other monsters didn’t even know what hit them.

Gaster, and the world he came from, is no more.

This part seems way too speculative, and not really hinted towards in any way that I can see.