Sorry I haven't a clue what kind of music you play, so I don't know if this will be of any help to you. I play Irish music, so maybe some elements of that might be applicable. When Irish musicians use left-hand chords, their reason for doing it is primarily to help accent certain notes in the melody. It's the same with cuts. You don't just cut any note - you want to be cutting to emphasise certain notes. It's the same principle with bass notes and chords, and that may be so for other types of music as well, apart from maybe songs where the harmonies you're creating with chords are probably more important than creating a danceable rhythm.
So if it is this that you want to achieve, then I would begin not by stressing about what harmonies you're going to use, but rather by going right back to the tune itself and deciding which notes in the tune are begging to be emphasised. A lot of the time you might decide that the emphasis belongs on the backbeat, but sometimes when there's a long note at the start of a bar, for example, it needs a chord or something to help emphasise it.
When you've decided which notes you wish to emphasise, then it's just a case of dropping in any bass note that will fit. A lot of the time a simple octave bass note sounds best, but sometimes others work better. Big fat growly chords often work on those long notes on the downbeat. If you mix up downbeat and backbeat emphasis, then you'll end up with a freer, more creative accompaniment. On the other hand, if you over-emphasise the downbeat, then you'll get a very steamrollered sound, and if you overemphasise the backbeat, you'll get a horrible, disjointed, reggae-like sound. If you emphasise both all the time, then you'll get that oompah effect that you don't want either. The trick is striking a balance, I think.
What concertina masters like Micheal O'Raghallaigh have done is taken this to its absolute limits. He is able to play all of a tune - or at least most of a tune - in unison octaves, so that he can then pick and choose freely which ones he wants to emphasise out of the whole set at his disposal.