Juristic discussions mention basmalah in relation to istinja, with positions about saying it before or after that act. If it is said before istinja, the appropriate time is before uncovering the private parts.
If the private parts are already uncovered, Allah’s name is remembered inwardly rather than pronounced. This reflects the etiquette of divine remembrance alongside the recommended opening basmalah of wudu.
The basmalah and istinja remain distinct practices with their own proper occasions.