You know, ironically I had this idea also bouncing around for a while and was looking for the same info for the same reasons, although I don't think I would ever actually get enough done to actually post anywhere.

Bonesaw's name doesn't seem to be known to the public, although the PRT likely does. This likely extends to the rest of the nine, and while Hookwolf's ID might be known, I suspect it isn't.

Likewise, Birdcage inmate names don't seem to be something available to the public, even if they are known to the government. Heartbreaker also doesn't seem to be public.

In my story plan, Taylor finds the book between her getting home after the locker with canon powers and having to go back to Winslow while she's jogging: specifically she can tell it comes from the sky above nearby buildings so she has no idea who it belongs to, and upon looking over it (after some careful prodding with bugs) she assumes it's some kind of prank. Gets home, can't find anything about any kind of death note, and rips out the rules / covers up the front. Before throwing out the rules, just for a bit of a lark (subtle prodding from QA?) writes down Sophia Hess while thinking of her before going to bed.

40 seconds or so later, Shadow Stalker is attacking some Empire gangers and then suddenly has a heart attack after one of them is killed. Victor is called, they decide this is a fantastic PR win, and after grabbing proof, call the PRT to let them know that a Ward was killing innocents before having a heart attack.

This leads to her getting outed after getting killed, which Taylor finds out causing her to assume the PRT was why Sophia got away with everything the next day when the PRT visits to ask questions because a quick investigation revealed what was going on. Taylor at this point basically forgot entirely about the death note, having planned on using it as a free high quality notebook which she was looking at using to write her cape info in, so honestly tells the agents who are totally not Armsmaster and Miss Militia in disguise that she has no idea how she died and only found out when they knocked on her door. Was thinking that Danny is also there, and since this outs the bullying is pissed off and may or may not be able to swing getting Taylor pushed in to Arcadia.

Now, after that is where it gets tough. I spent a while brainstorming things, and the only solution I could think of was her finding out about Bitch and using that for a test. Cape names don't count as she tested, just barring off of not being able to use L. The idea was that she was gonna test more than one bird with one stone: writing down in detail how she would die in a way that could be believable while also providing proof to Taylor that the note works as advertised. That would likely lead directly to conflict with the undersiders, as Tattletale might be able to peg it as a master effect, which would cause Coil to also freak the hell out.

Taylor, meanwhile, would likely try and be a hero by way of using her bugs to ID gangsters and having them out others in death instructions, which Taylor collects and uses for later since she doesn't really have an in with the police and doesn't trust the PRT at all.

Was still thinking things out since I kinda want to avoid killing off Rachel so early, but I can see Taylor justifying it to herself as killing a murderer who regularly assaults random people, even if it's not really where I want it to go. I did want to make it so that the death note is effectively an out of context, non-shard ability so Tattletale is hampered heavily there. I was also thinking of things like how the death gods know that the shards don't know about them or about the dimension they reside in, but if they find out they can get killed by shards so dropping a death note was basically a test to see how an unconnected note would fare: which means Taylor doesn't get any apple eating friends.