The 1st and 2nd should be the most crucial amendments. The 2nd should be the greatest defense against LCS victory, as once that's gone, the citizens lose their personal defense against LCS activism.
The 1st amendment on the other hand is the cornerstone of the liberal political agenda in this country. As soon as the 1st amendment is gone, the conservative agenda leaps forward because the 1st amendment was what banned C+ sodomy laws, what got abortion to L/L+, free speech obviously, flag burning and much more. It's the catch all excuse for the supreme court to make liberal decisions. If the 1st is struck down, a flurry of conservative law changes should follow.
Additionally, past this point, conservative media goes into overdrive, brainwashing the populace far more effectively. The liberal guardian represents the few underground liberal publications left at this point. Maybe this could be modeled by special "state press" media which doesn't follow the ordinary rules. 80% of the people who don't have a favorable view of the LCS are assumed to be getting their news from the state press. If these people don't hold conservative views, propaganda will quickly indoctrinate them. Every month, half the people reading the state press who don't hold conservative views on the issue give into the brainwashing. For simplicity, we assume that those who hold the LCS in high esteem have no conservative views, because they are a defiant counter culture. It's only once they become disillusioned with the LCS that they can be brainwashed.
So suppose at the end of the month, only 35% of the country holds the LCS in high regard. Of the remaining 65% of the country, 80% are watching the conservative propaganda networks, i.e. 52% are being brainwashed. So any views which are bellow 52% conservative will quickly deteriorate. Suppose CSS and LCS rampages had 80% of people wanting tighter gun laws. That's 45% of the country who supports a liberal view, but is only reading state propaganda about it. 80% of them are brainwashed each month, meaning in the first month, gun control deteriorates by 36% of the general population. This is all before the effects of other media, sleepers, etc comes into play.
Now think about what this does to the LCS's popularity. Society is being strongly tugged in the conservative direction every month. That makes the LCS less popular, which in turn makes the state propaganda reach more people. The country started at 65% conservative, about enough for amendments to start passing. But in one month, the state propaganda would have done huge damage to any liberal views held by society. Eventually, societies views will become roughly even across the board and state propaganda will stop working, there's no one tuning in anymore who isn't already brainwashed. At this point society is completely polarized into a state sponsored conservative culture and a liberal underground. But the game is probably over by the time the issues level out like that.
Additionally, any negative press against the LCS is deadly. Screw up at all, and the public turns against you and starts getting brainwashed.
So yeah, one mechanic, REALLY scary country...
It would be cool to make a tough endgame, but it might need some mitigating factors to make it less impossible. Maybe a fraction of those brainwashed become "disillusioned" instead? Disillusioned people can't be brainwashed into conservatism, but won't support liberalism either. However, if you survive and liberalism gets a big enough victory like the election of a liberal president, the disillusioned are inspired by Liberal hope and become liberals in droves, allowing a massive liberal reawakening, carrying you to victory.
This could also make for an interesting alternative start of game scenario. The game starts with society being left of center, a roughly balance but slightly C house and a moderate senate with say... one more liberal then conservative. However, the Supreme court has 4 C+ members and 2 C members and the C president just died in a terrorist attack, letting the C+ vice president succeed him. The new president used the crises to repeal the first amendment with only half of congress in quorum, which the court supports.
So basically, it's conservative semi-coup that's repealed the 1st and now society rapidly deteriorating into conservatism. The LCS starts the day after the 1st is repealed and needs to quickly become a national sensation and start publishing the Liberal Guardian to reverse conservative propaganda. This would make the middle of the game very hard, as society is very conservative, but make the start of the game safer as society is more liberal then before.
Such a scenario might benefit from the effectiveness of state propaganda depending on free speech law. At M, 20% brainwashing effectiveness, meaning 20% of those not reading the guardian succumb to brainwashing but there is still a free press to keep the brainwashing from being more effective then that. At C, brainwashing works 50% of the time as the government can now prevent free media from correcting their lies. At C+, free media is shut down so brainwashing shoots up to the full 80% and the disillusionment effect appears. Conservatives would naturally focus on overturning free speech first and foremost in a country without the 1st amendment.
Yeah, I kinda got carried away...
But basically four mechanics proposed for a country without the 1st amendment:
1) state media that works in a different way.
2) disillusionment
3) state media effectiveness scaled to free speech laws
4) conservative "focus" on overturning free speech laws without a first amendment.
And really, 1 is the important thing and everything else is just gravy.
I think the first 3 of these would be pretty simple, right? For 1 and 3, it's just another couple calculations on how public opinion changes at the end of the month. Concept 2 is a new thing to track, but doesn't need to show up anywhere except when calculating propaganda effectiveness and if the disillusioned become liberals in a liberal awakening, the other methods of tracking aren't affected and conservatism and liberalism still add up to 100% of the public opinion. Concept 4 seems like it requires the most new work. Maybe if there's no 1st amendment, the president could automatically issue an executive order in January of every year, bringing free speech law closer to his views if they don't match up?