We consider elliptic curves over global fields of positive characteristic with two distinct marked non- trivial rational points. Restricting to a certain subfamily of the universal one, we show that the average size of the 2-Selmer groups of these curves exists, in a natural sense, and equals 12. Along the way, we consider a map from these 2-Selmer groups to the moduli space of G-torsors over an algebraic curve, where G is isogenous to SL_2^4 , and show that the images of 2-Selmer elements under this map become equidistributed in the limit.