(Don't let your eyes glaze over on this one-- bear with me!)
If you strip this situation to its barest essence, what you're doing is trying to set up a game (in the game theory sense) within a game (in the role-playing sense.) What you want, specifically, is a co-operative game (theory sense) within the game (role-playing sense) rather than a competitive game (theory sense) which threatens to infect the game (role-playing sense) with an overly competitive edge.
To borrow a little bit of formalism from that game theory world, co-operative games have rewards and coalitions. Coalitions form when people in the game begin to work together because they can achieve greater rewards. What you want, formally, is for your players (their characters, actually) to bargain amongst themselves, explore a space of coalitions, and eventually settle on one.
(Yes, I am a nerd.)
But if all you do is set up a game-within-a-game where there is one big shared reward, and then a bunch of independent little rewards that apply only to individuals, and can be accomplished alone, then there is precisely zero incentive to form coalitions. There is nothing to be gained by Players 1 and 2 declaring alliance-- they don't share their rewards and they don't influence each others' success.
So, what can be done?
Well, you can have characters share goals, or share in each others' rewards. But without doing any formal analysis (I am not that much of a nerd) I think this is going to be unsatisfying-- the coalitions are probably in some sense going to be either obvious, or symmetric and unresolvable.
Or you can have the members of the coalitions affect the chances of success for the independent goals of the other members of the coalition. So for instance, Player A has some resource than can help Player B, and vice-versa. Or perhaps not vice-versa-- these coalitions can get very complicated very quickly. But now, there are incentives for co-operation, negotiation, etc.
Now, how do we put role-playing meat on all these abstract notions?
Well, one classic quest form is, "We couldn't kill the dragon without the magic sword, but to get the magic sword, the smith needed mithril from the Dwarves, who needed the Balrog killed, which was only vulnerable to..." It's a sort of a stack, where sub-quests keep getting pushed on the stack until a sufficient depth, and then the whole structure resolves.
Well, if the resources are within the group, this can become a similar structure, although a little more parallel: "Everybody wanted Joe to survive, but Tommy was working for his wife who wanted him to join her overseas. Well, that wasn't happening unless someone fixed his visa... luckily, Mack knew a guy who knew a guy. Or maybe Carlo could have him smuggled out. But Carlo was working for Joe's brother who needed for none of the charges to stick. Luckily, Mack knew a another guy who knew a judge...."
Even that's a little bit on the plot coupon side.
I would encourage you to throw a fair number of quirks into the arrangement-- maybe Mack knows a guy who knows a guy... but it's always the same guy, who can be used to support several goals in theory but only one in practice. Maybe Fast Eddie is an all around hustler who can help with a lot of things, but can't even do one of them all on his own. Etc.
And beyond that-- depending on how long you want this to play out-- I would encourage that when you're giving everyone these resources, that you don't be up front and obvious about how they can be used. Let the players explore that through role-playing. It may take some role-playing before Mack realizes that his guy can help out here, there, or the other place.
And finally, there is still always the chance that some player will somehow be locked out. Consider strongly the notion of "side payments" which is to say, some alternate form of reward (whatever in-game would be appropriate-- money, gear, favors from NPCs or PCs) to make these negotiations a little more fluid.