Apparently in EraMegatenP, the Factory dungeon (Rescue of the Shrine Princess request) is timed from when you accept the request? Pretty stupid when it unlocks day 1 or so and has lvl 30 enemies. Just spent a couple hours running around the dungeon trying to figure out what the hell I was missing to eventually give up and try fiddling with debug mode. From what I found, at some point between 3 and 8 days from when you accept the quest, the map changes slightly to include a "tower" with a locked door and at that point the priestess is too brainwashed to be saved.
(Of course it doesn't say that outright, it just says "She resisted being saved. If she won't regain her sanity, she won't move a spot from there" and doesn't mention there is no way to stop the brainwashing song from the tower - the locked door seems to just be scenery, with nothing actually connected to it. If you go back a number of days later, you can see at least two more levels of corruption, but of course can't save her.)
Posting this here to maybe save someone else the frustration in the future.
Yeah, EraMegaten is not afraid to seriously screw with you if you do something wrong in one questline since there are about a billion other ones. Having said that, you can still get to her, you just need to do the tower raid. According to the Japanese wiki, it should unlock when her brainwashing level hits 'severe,' whatever that means. So, probably a few days. This is because there's actually an ending locked behind leaving her in there for 45 days or so.
Having said that, I've just checked, and the average level of the factory ranges from 8-9 outside to 14-20 in the maze. There are a few bigger enemies, but they're gate guards. The one you're talking about, I'm guessing you unlocked all the doors in the B7 maze. That's a beaf gate to keep you from taking the shortcut route. Given you can re-enter dungeons pretty easily in P, I'd actually say rescuing her on D1 is possible, if very difficult, and rescuing her by day 2 is pretty simple if you know what you're doing and take advantage of requests and events.
You can apparently break the tower raid if you screw with the SMT I questline, so honestly I wouldn't worry too much on a first play. For context as to why: The 'factory' dungeon is a canon location from SMT II. The Princess is a new character, because the SMT II questline is both a hot mess and unfinished, but TL/DR, there's a character locked in the tower singing a mind control song. The tower raid request has one of the heroes, I... think? The chaos one, breaking in with an assault team to blow the place up. This is done in a request, not in the dungeon, which is why you can't get in the tower door. Once that request completes, the tower's brainwashing shuts down, and you can rescue the prisoner again. This is kiiiinda lame and obscure, but if you've played SMT II, you'll sorta get some of it. But in the end, yeah, EraMegatenP or Shin Era Tensei or whatever the name's been changed to can be opaque as hell, and having the wiki on speed dial helps. Sorta.
There's always an ending you can get, though, and failing that, you can loop after day 50 with a better understanding of things, some more stats on your PC, and maybe a few contracted favorites to help you out.
@Wabobo
Yeah, Eramaker is an entirely text based engine. Images are sorcery and a modification. The entire interface is made out of text so changing the font size without being careful can seriously screw with it. There's a reason the guide strongly suggests leaving it at 16.