You can try with finer discretization by making MaxCellMeasure smaller:

usol = NDSolveValue[{\!\(
\*SubsuperscriptBox[\(\[Del]\), \({x, y}\), \(2\)]\(u[x, y]\)\) + 
          0.5 Exp[-u[x, y]] - 2. Exp[u[x, y]] == 0., 
      DirichletCondition[
     u[x, y] == 2. Log[1. - 0.5 Sin[2. ArcTan[x, y]]],
         True]}, u, {x, y} \[Element] Disk[], 
   Method -> {FiniteElement, 
     MeshOptions -> {MaxCellMeasure -> 0.0001, 
       MeshElementType -> TriangleElement}}];

Eqsol[x_, y_] := Derivative[2, 0][usol][x, y] + Derivative[0, 2][usol][x, y] +
   0.5 Exp[-usol[x, y]] - 2. Exp[usol[x, y]];

Plot3D[Evaluate@Eqsol[x, y], {x, y} \[Element] Disk[], PlotRange -> Full, AxesLabel -> Automatic]

Furthermore, you can read in this Mathematica FEM Tutorial why this way of verifying the result of NDSolveValue results in an overestimated error.

You can try with finer discretization by making MaxCellMeasure smaller:

usol = NDSolveValue[{\!\(
\*SubsuperscriptBox[\(\[Del]\), \({x, y}\), \(2\)]\(u[x, y]\)\) + 
          0.5 Exp[-u[x, y]] - 2. Exp[u[x, y]] == 0., 
      DirichletCondition[
     u[x, y] == 2. Log[1. - 0.5 Sin[2. ArcTan[x, y]]],
         True]}, u, {x, y} \[Element] Disk[], 
   Method -> {FiniteElement, 
     MeshOptions -> {MaxCellMeasure -> 0.0001, 
       MeshElementType -> TriangleElement}}];

Eqsol[x_, y_] := Derivative[2, 0][usol][x, y] + Derivative[0, 2][usol][x, y] +
   0.5 Exp[-usol[x, y]] - 2. Exp[usol[x, y]];

Plot3D[Evaluate@Eqsol[x, y], {x, y} \[Element] Disk[], PlotRange -> Full, AxesLabel -> Automatic]

However, please read this Mathematica FEM Tutorial about why your way of verifying the result of NDSolveValue gives you falsely exaggerated error. It has to do with calculating the derivative of interpolating function, which is not numerically stable.

You can try with finer discretization by making MaxCellMeasure smaller:

usol = NDSolveValue[{\!\(
\*SubsuperscriptBox[\(\[Del]\), \({x, y}\), \(2\)]\(u[x, y]\)\) + 
          0.5 Exp[-u[x, y]] - 2. Exp[u[x, y]] == 0., 
      DirichletCondition[
     u[x, y] == 2. Log[1. - 0.5 Sin[2. ArcTan[x, y]]],
         True]}, u, {x, y} \[Element] Disk[], 
   Method -> {FiniteElement, 
     MeshOptions -> {MaxCellMeasure -> 0.0001, 
       MeshElementType -> TriangleElement}}];

Eqsol[x_, y_] := Derivative[2, 0][usol][x, y] + Derivative[0, 2][usol][x, y] +
   0.5 Exp[-usol[x, y]] - 2. Exp[usol[x, y]];

Plot3D[Evaluate@Eqsol[x, y], {x, y} \[Element] Disk[], PlotRange -> Full, AxesLabel -> Automatic]

You can try with finer discretization by making MaxCellMeasure smaller:

usol = NDSolveValue[{\!\(
\*SubsuperscriptBox[\(\[Del]\), \({x, y}\), \(2\)]\(u[x, y]\)\) + 
          0.5 Exp[-u[x, y]] - 2. Exp[u[x, y]] == 0., 
      DirichletCondition[
     u[x, y] == 2. Log[1. - 0.5 Sin[2. ArcTan[x, y]]],
         True]}, u, {x, y} \[Element] Disk[], 
   Method -> {FiniteElement, 
     MeshOptions -> {MaxCellMeasure -> 0.0001, 
       MeshElementType -> TriangleElement}}];

Eqsol[x_, y_] := Derivative[2, 0][usol][x, y] + Derivative[0, 2][usol][x, y] +
   0.5 Exp[-usol[x, y]] - 2. Exp[usol[x, y]];

Plot3D[Evaluate@Eqsol[x, y], {x, y} \[Element] Disk[], PlotRange -> Full, AxesLabel -> Automatic]

Furthermore, you can read in this Mathematica FEM Tutorial why this way of verifying the result of NDSolveValue results in an overestimated error.