To be truly resilient against any kind of (potentially unknown, even) lossy image compression, you would need to add a lot of redundancy. QR Codes are pretty much that, but their data needs to survive being “transmitted” through a blurry photo with arbitrary lighting, rotation and perspective, so they likely have a lot more redundancy than you realistically need. (EDIT: You might hit the limit where you need QR Code levels of efficiency relatively quickly, depending on the number of steps and how aggressive each recompression is.)

If saving and sharing can mean changing the image dimensions however, you of course need redundancy by that factor.

But yeah, that’s why we don’t choose lossy compression algorithms for binary data. If every bit counts, there is no entropy you can lose (though a lossless algorithm might find a more efficient representation, so to speak).