So, I've been pondering hard to try to see if I can come up with anything that could make this method work. It's not great, but I came up with something involving intents.
Upon activating the time machine, it notes the exact variables of the moment. It also "records" all memories.
When something goes back in time, it is allowed to make changes. However, the time machine will force all actions to lead to the exact variables as before, basically creating a "fate" for the moment of time machine activation.
However, the thing going back in time is able to change things based on intent alone. For example, the note would constitute as "written intent" for the mop to not burn.
So, for the mop to not burn, the time machine allows for the group to check their notes and build the time machine exactly as it would be had the mop burned, and then lead up to them sending a note back in time anyway, for whatever reason.
When the moment of activation arrives, everything is as it was, except for the intended changes. A few seconds later, the changes are "reconfigured" to the present, and the mop pops back into existence, and the original memories of the original activation are reinstilled.
Thus:
B, C, D send a note back in time, have Time 1 memories.
Time 2 happens, and the mop is not burned, but all the same variables are matched by time of activation. B, C, D have Time 2 memories.
Time 2's time machine is activated and Time 1 is officially caught up to. A few seconds later, Time 1 is reconfigured to have the intended changes, and B, C, D regain Time 1 memories, overwriting Time 2 memories.
Thus, if someone goes back in time, they must have direct intent to save character A. That can happen, and all variables will match except for chararacter A living. Character A will be aware everything is happening, but upon catching up to Time 1, those memories will vanish and she will have either a "dead zone" in memory or she will remember exactly right before she died.
This is difficult as well and creates its own problems, and also relies on a bullcrap "operates via intents" mechanic.
Furthermore, if the time machine leads all variables to be exactly what they were except for what you intend, then by virtue of that, you know you'll succeed because you are changing things, so there's no free will at all.
Egh. I don't like it.
All I need is SOMETHING to make it plausible, and it's looking like I'll just need to blow the damn thing up.
I don't know where the original plot was going with it, other than potentially bringing the character back to life, though it could have played a larger role in the future. But since it's so limited in its function-- or something-- it seems like it's more likely to be used once and that's it.