Sadly the only lossless way of rotating an image is to use an EXIF tag, not rotate the pixel content. But the forum isn't lossless (as previously stated) so it's not worth chasing that dream.
See my link, you can rotate many JPEGs losslessly, or all of them if you don't mind discarding a few pixels along the edges.
The forum can be made lossless by uploading the image in a ZIP archive, but that makes it a little harder to view
edit
And by the way, the forum's image "optimizer" plugin is sloppy and the attachment size that you see is the size of the file originally uploaded, not the one that you actually download from the server
This may make some people believe that they are downloading their original images, but they are not.