It is possible
The fact that David was accused of rape previously was definitely a circumstance used against him in court, as the judge and jury would see him as a serial rapist. But was the first "rape" part of the plan?
The alleged rape case has completely destroyed his life: he has lost his family and his career, and he became an alcoholic - those are tough things to go through, but possible for someone willing to die for an idea. But then I don't believe that he would come to this grand plan so early before knowing that Constance was terminally ill, which he didn't know until much later.
BUT there is the case of the postcard in the last scene: when his now ex-wife opens the suitcase, she finds the postcard sent by Berlin saying that it is "from a student who would do anything". Why would he send it to his wife? To show that he was innocent of the first rape? The court has said so and judging by the rather peaceful divorce, it seems that his then-wife believed that he had consensual affair (IMO, if you leave a cheating husband, you start with "trial separation"; If you think that you think that your husband is a rapist, you put a court order).
So there are two options here:
The first accusation at best gave David an idea, as it was showing that people are willing to judge and punish you without knowing all the facts, but he and Constance didn't plan the fake murder until much later until Constance's condition started to deteriorate.
David was lying to Bethany (and to the viewer as an unreliable narrator) and he decided to go with the plan much earlier and Berlin was at least partially involved in the plan.
Personally, I rather believe in the first version.