Exactly what it says. I just finished the Illegal Dumping quest, made the catalyst, went to see the Elf chief guy, but I don't have the manor open to go advance to chapter 6. The only item I have up under missions is to get promoted, and it's NG+, so all the shop requirements are already met. The quests I have open are:

1. Restore Yuela's sword
29. Clean Lake Shisetika
41. Crayl's Supplement
52. Promise with the Angel
64. Investigate the creature
65. Substitute Shells
85. Exploration
106. Spatial Rift
109. Shipwreck
112. Church Escort

I'm not sure what else there is that I need to do to advance. Any ideas?

No events in the city?
Are you sure you have both workshop upgrades and evaluation?
Try leaving the city and returning, or waiting a day.

The only event in the city is a bulletin board request for killing a monster, and the only character scenes in the shop are Wil's wigmaking and choosing Lily's form. I tried leaving the city and coming back, I went around and completed a few maps, but nothing is changing.

And it says they can't expand further, and my evaluation is at 24,873, so that shouldn't be the problem.

A story event is probably hidden under the bulletin board request. You need to take that quest, then you can progress with the story.
This should really be considered a bug.

Which monster do you need to kill for this quest? We can help with that.

Did you level all your characters? Did you transmigrate some characters and forget to level them? Might be you need a trip to Deadbeat.