Small nit: The lossiness is not a problem at all. Entropy coding turns an imperfect, lossy predictor into a lossless data compressor, and the better the predictor, the better the compression ratio. All Hutter Prize contestants anywhere near the top use it. The connection at a mathematical level is direct and straightforward enough that "bits per byte" is a common number used in benchmarking language models, despite the fact that they are generally not intended to be used for data compression.
The practical reason why a ChatGPT-based system won't be competing for the Hutter Prize is simply that it's a contest about compressing a 1GB file, and GPT-3's weights are both proprietary and take up hundreds of times more space than that.