Simultaneous equilibria
Ok.  You have five equations, two are differential equations and three are algebraic.  And you have the explicit assumption that the equilibria are fast enough so that the two equilibrium equations are always satisfied.

You also have five unknowns, A, B, C, AB, and BC.  One method of solution is to use the algebraic equations to eliminate three of the variables.  This should leave you with two differential equations with two unknowns.  If you are very lucky (I've not worked out the algebra) you will get two equations that can be integrated independently.  More likely you will have two simultaneous differential equations.

In either case the result is apt to be a mess.  So I'd cheat.  If this is a textbook problem or homework problem, there is probably a way to combine the equations to get simple enough expressions.  Otherwise, you may have serious troubles solving the set.