Three reasons exist why the go along with the Blackmail.
The most important of these is: It needs to be explicitly clear that if the blackmailer drops the photos and blog online, it goes viral and their parents will find out. Tim's dad is violently homophobic and would destroy his future. By agreeing to fake-date at school, they can actively control the flow of information. As they discuss in their "Rules of Engagement," they can pretend to be a couple at school, but lie to their parents by claiming they are just "study partners" and "stress relief partners". The fake dating is a firewall to keep the truth from reaching Tim's father.
Again I have not done a good job of making this clear in this chapter, something I might have been able to catch had I not been up all night hugging my toilet bowl and wishing the colour green did not apply to my face.
Hah, I will fix this logic CLEARLY in tomorrow's chapter.