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Apart from the suspension geometry changes from being too low and so long as you aren't banging your tire against the fender walls, your ride quality shouldn't change much from just lowering the car.
Ride height doesn't affect the shock travel length on the BC coil overs you have: ride height adjustability is independent from preload. It also happens that, due to these design choices, the shock body on theme is also shorter than factory damper body (leading to reduced travel length before bottoming out). This tends to lead to these types of systems requiring stiffer springs.
This is different from how budget (like the Tein Street Basis) and euro-style coilovers (like K&W V3) provide height adjustment. In these instances, when you adjust ride height, you also change the preload and available travel distance. However, many high end systems have very long damper bodies since they don't have to incorporate a separate ride height adjustment collar. Therefore, you can accommodate a softer spring and still avoid bottoming out.
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With that being said, make sure the preload is set-up correctly on your coilovers. Also ensure none of the shocks are blown. Other than that, all you can really do is change the dampening.
If those changes didn't help, then you know that the harsh ride is due to they way your coilover system was specified/designed. Some just ride harsher than others due to spring rate, dampening rates, allowable travel, etc.
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If you're looking to stay on a budget, Tein Flex A or Flex Z are really affordable and streetable entry level coil overs
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