Sorry I deleted your other message by accident (“delete” is right next to “reply”, oops!)
I like your idea about run-length encoding, base64, and gzip encoding and I briefly played with it but I don’t think I can bring the length down to something reasonable. E.g. Base64 encoding of the required 864 bits is still 144 characters long. But please let me know if you find some clever solution, I don’t know much about compression to be honest.