Thank you for your response. Ys I tried this but it did not work.

It is very difficult to interpret the solution formula, because there are so many ambiguities: inverse functions and noninteger powers, which have more branches in the complex plane.

It is a tough system. I would do away with InverseFunction this way, but it is stille hard:

oldSystem = eq1 && eq2 && eq3 && eq4;
invFuncInstances = Union[Cases[oldSystem,
     InverseFunction[_][_], All, Heads -> True]] /.
   InverseFunction[f_][x_] :> {InverseFunction[f][x] ->
      If[x === B1, g1, g2],
     f[If[x === B1, g1, g2]] == x};
{substs, newEqs} = Transpose[invFuncInstances];
newSystem = (oldSystem /. substs) && And @@ newEqs // FullSimplify
sol0 = Solve[newSystem[[{2, 4}]], {A, B}][[1]]
newSystem0 = FullSimplify[Rest@newSystem /. sol0]
FindRoot[newSystem0, {{g1, -3}, {A1, -1}, {B1, I}},
 WorkingPrecision -> 50]