This idea, taken to the extreme is mozjpeg. It is really advanced and can take advantage of a lot of cool tricks (like trellis optimization) in order to get the absolute best quality for the size.
It's called jpeg-recompress and is part of the jpeg-archive project on Github.
It currently can use mean-pixel-error, SSIM (the default), MS-SSIM and SmallFry (an implementation of the patented algorithm used by the JPEG mini app). So they may be open to adding this new comparison algorithm too.
edit: someone already filed an issue asking them to look at it: