Pure solo seems like a bit much, but I've tried modified solo runs. Only the Founder can go out and infiltrate locations usually (referred to as setting forth to fight EVIL or something, haven't played LCS in a while) while everyone else stays at home. Active LCS members other than the Founder are pretty much support, staying at home and selling pretty paintings, playing music for cash, tailoring new disguises and armor, stealing cars, selling drugs, recruiting captured prisoners, etc. The only time an active, non-Founding LCS member is allowed to venture forth out of the safehouse is for going to the pawnshop, car dealership, or department store to buy/sell stuff. Sleeper LCS members do sleeper agent stuff. In essence, it's kinda like a Liberal version of Peace Walker, especially because I love kidnapping and enlightening prisoners. I've never actually finished my Peace-Walker-Solo runs though. LCS is a very, very long game, even if you go full terrorist (which is super OP) and everything skyrockets to Liberal proportions.
The CCS can be defeated without fighting their stupidly overpowered bosses if you hack the CIA and have a Liberal Guardian. Going loud in a solo run is absolutely suicidal unless you've got insane Dodge, since no matter how badass your Liberal is, he'll only attack once per round... And so will the six guys outnumbering him. Do the math. Strangely enough, angry rednecks interrupting your Liberal Disobedience will cause your Liberal to singlehandedly beat the shit out of an angry mob if his Unarmed is high enough. That's the equivalent of a cutscene showing the hero doing really awesome stuff that will get you slaughtered during regular gameplay. Even with insane dodge, RNGesus will screw you over unless you savescum, since guns are harder to dodge than unarmed, and almost everyone has guns.
Theoretically, you could solo the first CCS safehouse at least by stealthing your way to the Boss, equipping a really powerful gun that you have high skill in, and hoping to cripple the boss in one hit. If his health goes down to Badly Wounded, he'll be stunned by the pain and unable to fight, making him easy to rip apart. I'd say an M4 Carbine (higher accuracy - better chance of headshot) or M249 (It's a machinegun with an accuracy bonus of 5 in source code due to spray and pray) would do the trick. Later CCS Safehouses do not allow you to stealth and force you to rambo your way through, which really sucks.