Yes I have tried this method but there is a problem at the numerator.
If we do this Simplify[expr, {n1 == n2, n1 + n2 == n}] we obtain
(n2 (2 A - 2 c + \[Gamma]1 + \[Gamma]2))/(b (1 + n1 + n2))
At the numerator n1 disappears
The Simplify function supports simplification with assumptions
Simplify
Simplify[expr,assum]
so you can do something like
Simplify[expr, {n1 == n2, n1 + n2 == n}]
