I don't know if I was actually clever or it was intentionally designed to be passed that way, but I found that walking backwards through the area on floor 4 that wouldn't let you pass lets you get by without triggering the checkpoint that pushes you back. The areas of floors 5 and 6 that you can access afterwards are just short hallways, and I hit another roadblock at floor 7, so here's my save up there if anyone doesn't want to deal with climbing up to it, themselves. At the very least, there appears to be more instances of the "big penis" enemies that I think have new animations up there, so there's that (though I've noticed that trying to trigger their animations with the blacksmith simply doesn't work - guess the dev hasn't created them yet). Floor 2 also has an area that walking backwards let me access, though it was just a group of succubus girls that you can talk to (and as with all of the NPCs here, I don't know if talking to them actually matters/triggers anything).
If anyone else finds more to do, let me know, and to those who are wondering how I got past the boss on floor 4, I just used Cheat Engine to lower its health. The heck with wasting precious time trying to grind/find an effective way to beat them normally; I'm already frustrated enough with the constant battles that I can almost never run away from, especially when many of the enemies seem like damage sponges that can hit the entire party and inflict status effects. I imagine the game would be a little less frustrating if I knew Japanese, though.
Edit/Update: I did a little more poking around and found that I could now open a door on the second floor that has at least two bosses behind it (a wandering dragon and some miko-looking woman), so there's that much more to do, at least. I'll update with more if I play much further and find anything.
Second Update: Beat those bosses (though I assume the dragon is optional/not important), found that the first floor door could be unlocked, beat the boss (and dragon) there, then unlocked the door on the fourth floor, now I can access the majority of the fifth floor. I'm surprised that there's this much to do in the demo.
Final Update: I'm just about played out at this point - I made it to floor 8 and there's two locked doors that I don't want to mess with right now (one on this floor, one on floor 6). I'm deleting my old save and attaching my most current one if anyone wants to take a whack at it. Oh, and there was a boss with 10,000 HP on the eighth floor that I happily cheated to kill.