@ez3
Sensei has agreed to pay for Rei for go to university in Tokyo. However, Rei's current plan is to die with Nagi and send Chako to Tokyo, so it's likely he's wants to use the money Sensei will give him to send Chako on her way.
Fairly certain he all but explicitly said that's his plan with how he promised to send Chako to Tokyo not long after Sensei told him about the money.
There's also the fact that Rei doesn't share anything that isn't relevant at that time - very typical behaviour of someone with his mindset, he mentions someone stopping him and Nagi, but not that it was Sensei and what happened after.
Also, Sensei's condition of being exclusive is practically Chekov's teacher-catching-Chako-and-Rei-looking-too-friendly-at-school waiting to go off. Seems like Sensei's current role is to be background noise until things go too well and she throws a spanner in the works, there's just too many avenues she could use to screw everything up.
#1. She catches Rei with Nagi. This could easily be seen as a "breach of contract" so she'd refuse to bankroll him, which means Chako doesn't go to Tokyo.
#2. She catches Rei with Chako, much more likely in my opinion. Once again, breach of contract but with the potential to trigger a bad end really quickly:
"Break up with her where I can see it or you're not getting the money." Rei, predictably, would break Chako's heart if it meant he could send her to Tokyo. Said heartbreak leads Chako right to Esemori as we've now seen that Chako goes into self-destruct mode pretty easily, and follows that exact course of action. Bad ending, with potential for worst ending depending on who is in the vicinity when that eventually goes down.
#3. Sensei gets her wish - a pregnancy, and somehow Nagi or Chako find out. I have a feeling Nagi - as someone who was left alone - might not approve of Rei dying if he's going to be a father. As for Chako, see #2.
#4. If Sensei feels her control of Rei is slipping, it'd be easy to schedule a home visit and apply pressure; she is his teacher after all.
All of this is ignoring the other pieces in play, such as Gen, Esemori's connection to Rei's mother, and what really went down at Lover's Abyss all those years ago.
God, I need to stop writing essays.