(This is too long for a comment.)
If you look at the result of
(Re[f[u + I v]] + I Im[f[u + I v]]) (\[DifferentialD] u + I \[DifferentialD] v) // Expand
I \[DifferentialD]u Im[f[u + I v]] - \[DifferentialD]v Im[f[u + I v]] +
\[DifferentialD]u Re[f[u + I v]] + I \[DifferentialD]v Re[f[u + I v]]
this gives a hint on how to compute $\Re\int f(z,\bar{z})\,\mathrm dz$:
parts = Simplify[ComplexExpand[ReIm[1/z^3 {(1 - Conjugate[z]^-4)/(1 + (z Conjugate[z])^-2)^2,
I (1 + Conjugate[z]^-4)/(1 + (z Conjugate[z])^-2)^2,
(2 Conjugate[z]^-2)/(1 + (z Conjugate[z])^-2)^2} /.
z -> u + I v], TargetFunctions -> {Re, Im}]]
{{(u (-1 + u^4 - 2 u^2 v^2 - 3 v^4))/(1 + u^4 + 2 u^2 v^2 + v^4)^2,
(v (-1 - 3 u^4 - 2 u^2 v^2 + v^4))/(1 + u^4 + 2 u^2 v^2 + v^4)^2},
{-((v (1 - 3 u^4 - 2 u^2 v^2 + v^4))/(1 + u^4 + 2 u^2 v^2 + v^4)^2),
(u + u^5 - 2 u^3 v^2 - 3 u v^4)/(1 + u^4 + 2 u^2 v^2 + v^4)^2},
{(2 u (u^2 + v^2))/(1 + u^4 + 2 u^2 v^2 + v^4)^2,
-((2 v (u^2 + v^2))/(1 + u^4 + 2 u^2 v^2 + v^4)^2)}}
FullSimplify[(MapThread[Integrate, {#, {u, v}}] & /@ parts) . {1, -1}]
{(-u^2 + v^2)/(1 + (u^2 + v^2)^2), -((2 u v)/(1 + (u^2 + v^2)^2)), -(1/(1 + (u^2 + v^2)^2))}
which you should then multiply by a factor of $\frac2{H}$.
Verify the constant curvature property:
Simplify[ResourceFunction["MeanCurvature"][2 {(-u^2 + v^2)/(1 + (u^2 + v^2)^2),
-((2 u v)/(1 + (u^2 + v^2)^2)),
-(1/(1 + (u^2 + v^2)^2))}, {u, v}],
u ∈ Reals && v ∈ Reals]
1
As an exercise, why is the following result expected?
GroebnerBasis[Thread[{x, y, z} == 2 {(-u^2 + v^2)/(1 + (u^2 + v^2)^2),
-((2 u v)/(1 + (u^2 + v^2)^2)),
-(1/(1 + (u^2 + v^2)^2))}],
{x, y, z}, {u, v}] // Simplify
{x^2 + y^2 + z (2 + z)}