It seems there is an undocumented parameter /hideconsole which will prevent the pop up to be appear.
The code looks like this now:
ExecDos::exec '"$OUTDIR\binary\SQLManagementStudio.exe" /hideconsole /QUIET /IACCEPTSQLSERVERLICENSETERMS /FEATURES=SSMS /ACTION=Install' "" "$OUTDIR\SQLManagementStudio.log"