It would average the same size as the actual data. Treating the pi bit sequence as random bits, and ignoring overlap effects, the probability that a given n bit sequence is the one you want is 1/2^n, so you need to try on average 2^n sequences to find the one you want, so the index to find it is typically of length n, up to some second order effects having to do with expectation of a log not being the log of an expectation.

You need both index and length, I guess. If concatenating both value is not enough to gain sufficient size shrink, you can always prefix a "number of times still needed to recursively de-index (repeat,start-point-index,size) concatenated triplets", and repeat until you match a desired size or lower.

I don’t know if there would be any logical issue with this approach. The only logistical difficulty I can figure out is computing enough decimals and search the pattern in it, but I guess that such a voluminous pre-computed approximation can greatly help.