This depends. If you're arguing the police are incompetent at protecting citizens from criminals (which is, all other things being equal, what you're doing simply by not being caught if public option isn't positive towards the LCS), then yes, self-defense arguments are more applicable. If you're arguing that the police are brutal, unaccountable, uncontrolled loose cannons (i.e., publicly funded vigilantes pursuing their own agenda as they see fit), arguing that the proper response is more vigilantism would be a harder sell. The trick is to not "beat the police", but to "rip the corrupt heart out of the fascist goon squad so it can be replaced with accountable elected peacekeepers". Spin, and all that. Though yes, if public opinion on police is on the C+ end of the spectrum, I'd wholeheartedly agree that defeating the police at that time would have undesired side effects.
OK, I have never even been in the USA, so I'm not a correct person to argue in that matter. From what I heard, many Americans view the second amendment not only as a mean to protect themselves from random violence, but also from excessive government control. That's why I assumed that telling the people that the police is not to be trusted would result in people arming themselves.
OK, another random idea - organisations could be used to add some crime syndicates (the mob for example). They wouldn't try to change the law (at least by public protests), but they could:
1. Inspire fear in the population - their actions would always be seen as rampage. They are a good reason to respect the police and demand better weapon access.
2. Actually help the player - we already have the black market. I suppose that it should be controlled by one of the crime organizations. That's it, you pissed off Mr Corleone, no M16s for you.
3. "Patron" over some activities, especially illegal ones.
Animal Research - Obviously, just putting them in the news would raise interest.
Especially if you let some misshapen nightmares loose.
Death Penalty - Get it in the news, probably by having a member of yours martyred.
Police Behavior - Get them to raid you with casualties.
This could be effective when most of the casualties never did anything more outrageous than spraying graffiti. Or, even worse, they're well known from their volontary public service. Killing the "Dallas Butcher" would, in fact, encourage people to support death penalty.
Homosexual Tolerance
I'm not even sure that it should exist as a separate issue. It usually comes in a package with other, sexually-themed opinions. For example that prostititution shouldn't be penalized, condoms are not for filthy perverts who want to engage in adultery without consequences, and so. Of course an individual doesn't have to support every sexual activity or none, but they could be grouped by amount of liberalism nessesary to accept this. Most of the people who view pre-marital sex as sin would propably be opposed to homosexuality as well.
I believe that an answer to the question - should the homosexual tolerance be a separate issue? - depends from what opposition should be implemented in the game. If we assume that it's all about the religion, more generalized sexual-themed issue would be just fine. We could add a church as an organisation and/or a site and that should be OK. If we want implement, for example, feminists (who have nothing against homosexuality, but are ideologically opposed to prostitution or porn industry), at least those issues should be separated.