Thank you for this series of posts about johnson distributions.
About the CDF, I've found this formula to calculate the CDF, using the builtin erf function (inspired by the documentation for Johnson distribution on Wolfram documentation:
Below my code, where the macrovar, are calculated using proc univariate

data SU_cdf;
do x=&theta-4*&sigma to &theta+4*&sigma by 0.0001;
cdf = .5 * ( 1 + erf((&Gamma + &Delta*arsinh((x - &Theta)/&Sigma)) / sqrt(2)));
output;
end;

This approach gives me good results. In my test case.
Thanks