You can't coup a city state that has DOWed you. The chances of a successful coup drop exponentially with increasing influence gap between you and the current ally.
You're not in a terrible position, though - I sometimes deliberately DOW the world for a tactical challenge, and everything usually goes fine, especially on tiny islands. Just remember the principle of force concentration - attack each enemy city with roughly 15+4N units one at a time (assuming equal tech footing), where N is the number of defenders. This is especially easy to do on Tiny islands, once you have an ocean-going navy. Concentrate your fire on surrounding defenders before attacking the city itself, and do not bother to save wounded units - a wounded unit draws enemy fire just as well as a healthy one, and you need your healthy units to remain healthy in order to deal more damage. Be selective and try to take only capitals of civs which have lots of wonders (these capitals are usually wonder hubs) if happiness is low.
From what I see and assume from your post, you have 3 serious problems, however. First and foremost, the "leading AI" has not been dragged into the war. As long as this is the case, you are losing the game. DOW that AI as soon as possible (as soon as you have the troops to deal with their attacks, that is) and make them pay. You don't want to be RAing or trading with that AI anyway. Even if you can't take its cities, tying up some of its production in unit building, by killing a few of its attacking units, gives you a better chance of winning than letting it tech up in leisure as the rest of the world burns. Secondly, expect to switch ideologies later in the game, since it is very likely that your ideology which will be established 2 turns later will not be the world ideology, and you will suffer a revolutionary wave as a result. Make sure to switch ideologies as soon as it is no longer practical to hold on to your original one, and to plan ahead for this eventuality. Thirdly, your unit composition is not well-suited for this situation - the rule of thumb for number of units required for attaking cities given above refers to an army or navy composed almost entirely of ranged units, with only a few speedy melee units to move in for the kill once the city has been reduced to 0 health. In your case, your galleasses are not ocean-going units, and therefore you will have a lot of trouble concentrating them on one city, even if it is possible to concentrate them at all. NEVER attempt to take a non-peninsular/island city with massed privateers, or even with massed ironclads or destroyers; the shoreline will restrict you to attack with only 2-3 units at a time, your units will be below half health after each attack, ground-based ranged units will be firing on those wounded units in addition to the city, you won't be able to reply to that fire at all, and the city will usually be recaptured a few times since you could not deal with the surrounding units first. Casualties will consequently be insane. My advice for your position would be to build new frigates in every city that can, and to delete any sufficiently inexperienced galleasses to free up the upkeep, if you can find it in your heart to do so. You may be able to upgrade the most experienced ones eventually if you free up enough gpt.
Keep hitting the enemy harder than the enemy is hitting you, and you will be fine. Nuke indiscriminately if you are the first to nukes. Do not even try to negotiate peace if you are not willing to cede a city or two to each AI - the AIs will come to you for peace once you have given them a good thrashing, and will refuse to give you a dignified peace should you fail to do so. When they do give in, just remember to emphasize gpt reparations and luxuries over cities, unless you have an abnormally happy empire by then. It may also help to listen to "Rule Britannia" on an MP4 player as you play.
