Is there any particular style that you're drawn to?
As the others have said, Bals are renowned for their smooshy slouchy styles but some styles are definitely more slouchy than others. I've never read of or seen anyone using a base shaper with one but I feel like it would look quite strange.
For a slightly less slouchy style I'd personally go with the Midday - its a hand-held bag though and for some people that's a bit of a turn off but its also a very ladylike design which I guess is what you're usually drawn to if your collection is mainly structured LVs and Chanel. Its also reasonably large - to get a better idea, you might want to have a look at the Midday clubhouse where loads of ladies have mod pics with theirs. The other problem with the Midday is that it is a discontinued style so you may need to go down the preloved route - that being said boutiques often carry residual stock from previous years (this is how I got a brand new Midday after it had been officially discontinued) so you could always try your luck with that.
And coming full circle, I like and have a lot of structured LVs and Chanels too but I go to Bal for something completely different and I would recommend a quintessential Bal to anyone - they are far more versatile than people often think and as you said before, they're just too darn gorgeous! The best idea would be to get yourself down to a store (if there's one near you) and play around with different styles. That way you can really see what will make your heart skip a beat. GLD!