The problem is that LR does not export to a "size on disc" which is (I think) the size you are looking for.
The export dialogue mentions

RESIZE TO FIT - and various ways of looking at the size, all of which are to do with the image size or dimensions in pixels, not the size on disc.

The size on disc is controlled by the quality setting (jpegs) and no two jpeg images will be the same. The simplest jpg will be an image of a black cat in a coal cellar and will yield a different number of bytes on disc to that of an image of a multicolored cat in a flower greenhouse.

In LR, uncheck the box "Resize to fit", and export the same jpeg two or three times at different quality settings. (the dpi - which is actually ppi - does not make a difference to the file or image size).

Which trick? The log out and in or the direct download?

Full disclosure: In my secret internet life I have been anointed with the title of "Adobe Community Professional" or "ACP". It came because I was recognized by Adobe as being able to write answers on their forums. In return they give me free software.

It is important to me that you understand that you DO NOT need to abandon LR6 if you buy a new camera. Yes, I think the CC plan is a good bargain. Adobe would like you to subscribe. But Adobe also works at providing tools to keep you running with what you have.

The "DNG Converter" is kept current with all the software that works with RAW from all the cameras that shoot it. You may not have used it correctly. But when you buy the A7Riv next year, you can use the DNG Converter to copy and convert your files to a RAW format that your then "old LR6" will do fine with. Adobe moves foward, but they don't force anybody to do anything.

People are buying new cameras all the time and still using their old favorite versions of Adobe software.

"I had already tried using DNG but to no avail." That's because "DNG Converter" is a separate, independent program. You use it to convert your Sony files to a different, more compatable, still RAW, format with a .DNG file name.

There are several things to try. First, try logging out and back into your Adobe account via the Help menu. Then try the Help>Update choice.

If that doesn't work there are a couple other things to try, including a direct download: http://swupdl.adobe.com/updates/oobe/aam20/win/AdobeLightroom-6.0/6.14/setup.zip

That link comes from here: https://helpx.adobe.com/lightroom/kb/keeping-lightroom-up-to-date.html
...and here: https://helpx.adobe.com/lightroom/kb/lightroom-downloads.html

If you still can't get it to work, there are two other forums that soley focus on LR user to user help.