Do you want the "lightest" "lowest tension" strings, then buy a 5-string set of E-C, dump the E string and put the A string in the E place, etc. In the double bass world on the other side of this forum, this is called "bumping." You will get an effective set that has about 20 pounds of tension per string. BUT...you will have no tone, no consistency, no stability in tuning, and everything will pull sharp on the frets.
I just want to comment on this part, because I have 2 basses set up like this, different reasons, and I only partially agree with your description. The first bass is a Fender Jazz with a snapped truss rod. You're right that there is a loss of consistency, some loss of stability in tuning, and things do pull sharp, especially at the deepest part of the neck bow. But this would be the case - and actually worse - with a heavier gauge set of strings, because those issues are caused by the truss rod factor, and not the strings.
My other bass is a Modulus with a graphite reinforced neck, which is dead straight with or without tension on the neck. I decided to try the detuned set on this bass exactly because it was so straight, and would be able to provide consistent action with either a heavy or light gauge set of strings. And my experience has been excellent since doing this, somewhere around 2 years ago. I use this bass frequently, and the tone is excellent - in fact it is my preferred bass for most situations. No issues with tuning at all anywhere on the neck, and this is a 24-fret neck.
In the end, from my experience if you have a neck that isn't unduly affected by lightening the tension, this method isn't necessarily going to cause any sorts of problems. If your neck isn't very straight however (even when adjusted to accommodate whatever tension is on it), or like the OP it cannot be straightened, then you will definitely have some problems - but those are mostly a function of the neck and not the strings per se.