That would add noise to the information the attacker gets, but ultimately the "output" length has to be correlated with the input length, so you can't ever get the amount of information the attacker gets down to zero which is what's needed to make a cipher secure.
'ultimately the "output" length has to be correlated with the input length'
But perhaps not relative to the size of the secret data, right?
(Note that I'm not saying, "Oh, obviously we should just do this to avoid that attack" - I'm hoping to learn by carefully understanding where it breaks down).
CRIME works because the compression/encryption algorithms aren't aware of which data is secret and which isn't. They treat the whole HTTP stream as a stream of text and operate on that. To them, text within the page is just the same as cookie data in the corresponding header field or (for example) the URI in the header.
If such a distinction were possible, the system could just not compress secret data. But if that were the case, nonsecret data wouldn't even need to be encrypted; it's nonsecret after all.
