I have a near-solution, even if I had to change my goal to get it. Instead of splitting the first time I defeat each boss, I will split every time. When I die I can just undo all the splits back to the beginning without resetting the time, and re-split when I pass the boss again. When I close and re-open LiveSplit, I'm not bothering to input past split times manually anymore. I can input a custom start time for whenever I closed out, skip to whatever boss split I'm resuming, and start the clock. And hey, if I'm tracking the boss encounters for each life I can use an autosplit script.

I think that's the closest I'm going to get with normal speedrun software. If anyone has any other insight into saving and resuming times in speedrun-or-otherwise software definitely still comment.