An even more 'classic' piano voicing might be LH 'shell' (3rd and 7th), melodic stuff in RH and leave the bass to the bass player. But that's not the whole answer either.
Remember, the inversion is determined by the bass note - whoever's playing it! If there's just a piano, your main responsibility is to provide the bass line. YOU decide what inversion the harmony takes. If there is a bass, your job is to keep out of his way and do whatever is useful, rhythmically, harmonically and melodically, with either hand, you won't be playing in the range that determines the inversion. If there's a guitar as well you've got someone else's toes not to tread on (and he on yours, though it can be hard to stop a guitarist strumming along :-).
Replying to your comment - for SOLO piano, your RH will generally want to play the tune, whether a tasteful rendition of the original melody or your own improvisation! There are various techniques of adding the harmony 'in the middle', a stride LH, block chord voicings under the melody in the RH, sustained 'thumb notes'... We're (presumably) not improvising Bach-style choral harmony, so we don't have to worry too much about doubled major 3rds, parallel 5ths etc.' although, as in all music, they WILL stand out aurally.