Selective format-compliant JPEG encryption?
While I haven't heard of previous work on this exact problem,
splicing together some standard solutions to a few other problems almost solves this problem.

There is a loophole in the pigeonhole principle:
if the output file is bigger than the input image,
it is possible to store "extra" information.

Starting from the blurred image you want everyone to see,
you somehow want to add more information to it so that the appropriate people can extract an unblurred image.

picture frame

Perhaps you could keep every pixel of that blurred image unmodified, and add something like a picture frame above, below, or all the way around the image to store extra data,
like the way dotless QR codes store extra data without modifying the center parts of the image.
There's been a lot of work on designing barcodes in such a way that it's possible to recover all the original data, even when the barcode is distorted far worse than simply uncompressing a JPEG and then recompressing with worse quality settings.
If you have a very wide bezel picture frame, it may be possible to encode (an encrypted version of) the entire (unblurred) original image file in the frame in the form of a standard barcode.
If you create a "residual image", the difference between the (unblurred) original JPEG file vs the blurred JPEG image in the middle of the frame, that residual image should be all-zeros except for the regions that were blurred out, and so (a compressed and then encrypted version of) the residual image should require far fewer bits to encode than the entire original image, and therefore require a narrower frame around the blurred image.

borderless

If you want steganography and you think the frame is too much of a giveaway, there seems to be a bunch of research on digital watermarking designed to be robust enough to survive uncompressing a JPEG, making minor modifications to it, and recompressing.
Perhaps you can tweak such watermarking and steganography techniques enough to store (a compressed and then encrypted version of) the residual image.
"Hide data in files with easy steganography tools";
"How can you hide information inside a jpg or gif photo?";
etc.
If that doesn't give you enough storage space to store the entire residual image, perhaps you could scale up the image (each pixel of the blurred original image expands to 4 pixels of the expanded image),
giving more area to encode data.
Or perhaps you could scale up the blurred image, and then apply something like DataGlyph halftoning to create an image that from a distance looks like your blurred JPEG file, but the Dataglyph digital halftoning contains (a compressed and then encrypted version of) the residual file or the entire (unblurred) original file.