Is it correct to say "Bismillah" (in the name of Allah) and "Al-Basmalah" (the full "Bismillah Al-Rahman Al-Rahim") when performing Istinja' (washing after defecating or urinating), and is the doer sinful if it is not correct?

Naming (saying "Bismillah") when cleaning oneself after relieving nature (istinja') is not from the Sunnah. It is disliked (makrooh) during the act of relieving oneself or inside the bathroom, because it is a form of remembrance (dhikr) that is prohibited at that time. Believing in its permissibility is an innovation (bid'ah). As for istinja' itself, it does not require intention (niyyah), because purification from impurity does not require intention according to the jurists.