Soft bits in large constellations
It depends on your decoder; for example, in message-passing decoders, most commonly, a prefactor doesn't affect the results at all, so you can scale to your heart's delight.
It's also common to just clip – whether a bit has a LLR of -100 or -90 doesn't really matter, does it.

However, the actual confidence levels don't make sense then anymore.

Well, you'd have to transform them, too. Notice that scaling LLRs makes sense in a Gaussian noise setting, as LLR involves taking the logarithm of a $ e^{-a|x-x_i|^2}$ function, so you're left with something rather sensible (scaling the LLR just scales $a$, which is common to all samples and constellation points). I don't know how your noise looks or how you arrive at confidence levels, but try to apply the logarithm to a ratio of confidences and see whether scaling the LLR preserves sensibility.