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]