Why can’t we know the specific outcome of an exchange economy with initial resource endowment? In a 2-person exchange economy we define the contract curve as the subset of all Pareto-efficient outcomes within the ‘lens’ bounded by both agents’ indifference curves through $\vec{R}$ which is the initial endowment vector. It is claimed that we cannot know which point on the contract curve is the actual outcome, as it depends on each of the agents’ bargaining skills. I read this on the Wikipedia page about Contract curves.

But isn’t the solution to the optimisation problem a unique price ratio $\frac{p_1}{p_2}$ (assuming convexity of preferences, local non-satiation and WARP), giving rise to a unique line in the Edgeworth box as each of the consumers’ “budget constraint”, and hence a unique equilibrium which is the intersection of this line and the contract curve?
The idea of the contract curve is just restricting the Pareto set so that no one player is worse off than the initial allocation. There's no concept of "price" involved here. When your endowment is $(x, y)$ and you're asked whether you'd like to get another bundle, $(x', y')$ instead, you only check if your utility is higher. If it is, then you'd take it.

The concept of price comes in with the Walrasian equilibrium (p. 654 of MWG) which is yet another level of refinement on the contract curve (i.e. the "core").