Assalaamou'alaikoum.
If you have had sexual intercourse, it is compulsory for you to take a ghusl before the next prayer.
However, it is sunnah to perform at least wudhu before you go back to sleep after sexual intercourse.
If you are already taking a body bath, then i would advice just to have the intention of ghusl by just wash your private part, perform wudhu and then wash the whole body and it will be counted as Ghusl.
it's the same what you've mentioned of just body bath.
Allah knows best