@Pauly51 Hartmann would definitely hand the marauder’s map in to the teachers 
@Bill you’re in luck, here’s a huge step by step guide for what you’re after. (I’m going to spoiler tag your post just so someone’s less likely to stumble on it)
MASSIVE, HUGE SPOILERS FOR EVERYTHING BELOW
This is a complicated one so buckle up for a huge essay! There’s a hidden stat called “backing” which describes how successful you are when confronting, or secretly taking down, Lady Renaldt. There are various ways you can increase it, and this is the most reliable way I think.
- Always go straight to Karson after getting the note.
-#I wonder if I could benefit from this (Karson relationship)
-#We should blow the whistle (Virtue)
-#I have enough friends who will rally behind us (Popularity)
-#We should talk to Mr Griffith (Griffith relationship)
-#We should talk to [the teacher who wasn’t fired] (that teacher relationship)
Then be a double agent, pretending to work for Lady Renaldt:
- When you go to Karson’s cabin:
-#I’m doing it to take Lady Renaldt down (Karson relationship)
-#She hates me ( low Renaldt relationship)
-#Don’t moralize at me (Virtue) - If writing to Inspector Burgin to prime her before graduation:
-# use my manipulative skills to butter her up (Manipulative)
-#show how much I share the college’s traditional values (Traditional)
-#lay it out without flourishes (Direct) - If visiting Mrs Benton:
-#Tell her directly (Direct)
-#Tell her gently (Manipulative)
-#Tell her angrily (Domineering) - If meeting with Annick Leuzinger:
-#Tell her directly (Direct)
-#Tell her angrily (Domineering)
-#This isn’t what Gallatin should be about (Traditional) - When working to get Blaise out:
-#Talk my way into collecting blueprints (Manipulative)
-#Calculate timing (Wit)
-#Pick up supplies for them (Poise) - When Blaise arrives:
-#Remind them we’re on the same side (Flair)
-#Let Blaise grab me (Accommodating)
-#Show Blaise who’s boss (Domineering)
Talk to Mr Griffith
- Initial reveal
-#I have a reliable reputation (Virtue)
-#I’ll appeal to the rapport (Mr Griffith relationship)
-#I’ve gotten to know [non-fired teacher] (teacher relationship)
Next it’s your preference whether you want to go public or deal with it more subtly.
- Speak to Inspector Burgin
-#I was an innocent / I had no choice (Virtue)
-#lean on my excellent grades / my schooling is important to me (Grades)
-#ask Karson/Blaise to speak up for me (relationship)
-#ask [classmate you sent to the mine] to speak for me (classmate relationship) - Confront Lady Renaldt individually
-#Inspector Burgin will listen to me / no one could think I’d do something dreadful (Virtue)
-#My friends will follow me, and won’t listen to you (Popularity)
-#My [clique] will back me up (clique strength)
-#your guests care about grades, they’ll believe me (grades)
-#We both stand to gain from this / I don’t believe you’d do that to me (Renaldt relationship)
-If you’re romantic with Auguste: how can you hurt Auguste? (Auguste relationship)
-If you’re romantic with Rosario: Rosario will protect me (engaged or Rosario relationship)
After accusing Lady R privately, she may back down if your backing score is high enough and you can push her to resign. Otherwise you’ll need to accuse her publicly.
- Accusing Lady Renaldt publicly (whether instead of or after accusing privately)
-#Call upon your clique (clique strength)
-#Trust that my friends will stand up for me (Popularity)
-#Call for [non-fired teacher]/Karson/Blaise to stand up for me (relationship)
-#lean on my trustworthy reputation (Virtue)
-#why would I lie about this when I’ve worked so hard? (Grades)
-#appeal to Beatrix [if you have the internship] (Virtue or Popularity)
-#appeal to the Chancellor [if you’re going to Gessner] (grades or Virtue)
-#appeal to Jerome [if you have the internship] (Virtue or Popularity)
Then in Chapter 12, depending on the success during the graduation scene, if you’ve revealed Lady Renaldt’s scheme to anyone the case will be in a better or a worse state.
If you weren’t so successful in Chapter 11, you can:
-#write an impassioned plea for justice (flair)
-#record meticulous details (wit)
-#I’m furious about this (Domineering)
-#Write whatever I think the Board needs to hear (Manipulative)
and there’s a chance that you’ll be able to have Lady R removed at that point. If you were successful in Chapter 11, she’ll be removed and potentially imprisoned.