You are mixing up two different ideas. You said that making Lin's dungeon story event independent would cause problems. I agreed with you since you are the author (don't change her story or make the event independent).
"...That's how it already works..."
Not true. You can invite Lin to the dungeon but if you ask her to do anything she exits with a "Friendship is too low" message.
1. Invite Lin to the Dungeon
2. Lin - "Yes Master"
3. Player - "We're going to have some fun here"
4. Lin - "I can't do that"
5. Game message - "Lin's Friendship is too low"
Summary of what I said below - Don't allow Lin to be invited to the dungeon until she completes her dungeon story event. Don't allow any girl to be invited to the dungeon unless they meet the requirements to do the most basic event in the dungeon. Add additional logic to all the girls in the hotel to accomplish this (read more below).
The best solution (without changing Lin's story) would be to:
"If it breaks the story like you suggest then there is a simple solution for all the girls in the hotel - add additional logic so that no girl can be invited into the dungeon unless they meet the minimum requirements to do at least one event there.
i.e. unless Lin has done her dungeon story event introduction then she can't be invited to the dungeon. Adding logic to all the girls would make sense here. If they don't meet the minimum requirements to do the most basic event in the dungeon then they most likely wouldn't agree to come to the dungeon in the first place. Problem solved. We just solved both of our concerns."