Primogeniture being the rule of succession pretty much rules out saying "not Henry".
And civil war is not going to help matters.
If Henry's the mess he seems to have been, I doubt he'd have the followers for such a war, or at least, not for long as his insanity becomes more apparent.
And this is the middle ages.
Henry, when his 'troubles' first become evident, could easily have one of those famous 'hunting accidents' that so many crowned heads of the era ran into, from time to time, couldn't he?
If Henry looks like a disaster from the outset, would you really put it past the men in high places of the time to 'solve' the primogeniture problem before Henry can destroy their fortunes?
Wouldn't be the first time in history that an heir apparent was eliminated by people who stood to lose everything should he/she ever get their hands on power, would it?