If we use John Rawl's concept of justice based on "A Theory of Justice" on the organisation/institution of Fair Trade, what would our results be? Would Rawls perceive Fair Trade as a fair/just organization because it helps the most disadvantaged people (the producers)? Which principles are represented and which are missing?
Fair Trade and Rawls Theory of Justice can be more broadly subsumed under the notion of Distributive Justice.
However the former is a practical step within existing institutions and the global economy to obtain a fairer deal for the disadvantaged, it is not a political philosophy; whilst the latter is a political philosophy that aims to
provide moral guidance for the choices that each society faces right now. So, for instance, advocates of Rawls’ Difference Principle are most constructively understood as arguing for changes to our basic institutional structures which would improve the lifetime prospects of the least advantaged in society.
This is very different.
Would Rawls accept fair trade as a fair/just organization?
Thus I would suggest, that a Rawlsian whilst commending a fair-trade initiative as ameliorating the lot of the poor, he/she would not think it Rawlsian as it leaves untouched the basic institutional structures that promote and sustain inequality.
Not much happens when you apply Rawls to your question. Rawls two principles are:
1)Liberty Principle: All should have the same basic freedoms (e.g. of association, expression, conscience).
2-a) Difference Principle: Social inequalities, if any, should be arranged so that they are to the greatest benefit of the less well-off.
2-b) Equal opportunity principle: Social inequalities, if any, should be attached to positions that are open to all in conditions of equality of opportunity.
It is not obvious that one can connect Fair Trade to the Theory of Justice, at least without some acrobatics, since Rawls's theory is about how to distribute goods in society in a just way. Maybe if you squint you can see Fair Trade as protecting the Liberty Principle by getting rid of forced labour, or enforcing the Equal Opportunity principle by enforcing equality among producers, but that is forcing it. Fair Trade is about ensuring standards of production that benefit the producers avoiding unethical relations of production, not about designing a system of distributive justice.
Rawls is not the theorist you are looking for. Nor are most of the Luck Egalitarians (like Dworkin, for example), since they all seek equality and justice through the analysis of individual desert. Fair Trade doesn't care if you are worse off by choice or accident: it combats unfair and exploitative labor relations even when they start from consensual contracts.
Since Fair Trade is about addressing inequality in relationships, you need a theory that focuses on equality and justice in relationships (in your case relationships of production and consumption). G.H. Cohen has some ideas about justice that depend on being able to participate in equal terms of collective production, which may be closer to what you are looking but not right there yet.
However, the two people you should definitely look for if you want to have an analysis of justice and the Fair Trade organisation are Iris Young and Elisabeth Anderson. Iris Young has written on how injustice is a result of unequal relationships that expose people to (a) marginalisation, (b) status hierarchy, (c) domination, (d) exploitation, and (e) cultural imperialism. Fair Trade purports to solve some of these. Young also writes about how consumers hold collective responsibility for sustaining a system that exposes people to these forms of unequal (she says oppressive) relationships.
Elisabeth Anderson has an entire theory of Democratic Equality that is focused on relationships that enable people to function on equal terms as members of a democratic society. That theory holds that everyone has inalienable rights to a sufficient level of capabilities to participate as an equal citizen. Since much of what Fair Trade tries to solve is about unfair labour relations, it is actually enforcing Democratic Equality by forbidding the kind of contractual relationship that would keep people from functioning as an equal citizen in a democratic society.
So to sum up, Rawls doesn't do much for you; luck egalitarians don't help either; you need to go for some pragmatists who address justice in (labour) relationships.
[Edit: Fair Trade may also be seen as enforcing Nozick's notion of self-ownership (which is a theory that focuses on relationships if put that way) by the producers insofar as it prevents slavery and forced labour, but you don't really want to go there. Nozick would say that ripping off producers is just so long as it is mutually consensual, and that Fair Trade's interference on these contracts (when the contract is consensual) is a violation of the right to self-ownership by the parts involved in the contract.]