1) Download a program like DirLinker, or learn to use symbolic linking another way (command prompt)
2) Get Dropbox, or another program that can share a folder across multiple computers
3) Run Program from 1
4) In 'Link Location', type (or use the folder to find) the location of your Hotkey settings folder (mine is at C:\Users\<USERNAME>\Documents\My Games\FINAL FANTASY XIV - A Realm Reborn)
5) In 'Link To', type (or find using the folder) your Dropbox or other programs sharing location (for me, D:\Users\<USERNAME>\Dropbox\Public).
6) Make sure that under 'If the link location exists', that 'Copy Contents to target then delete it' is selected
6) Click 'Go!'.
7) Do the above on the 2nd (and any additional) computers.


What this does:
DirLinker will create a symbolic link at the normal folders location, redirecting anything that saves to that location (FFXIV) to the Dropbox location. Dropbox will then upload the changed files, which the additional computers will then download and overwrite their copies of those files. This will cause any changed to Hotbars to be saved across all of the computers when they are changed on any of them.

Note:
This will not cause your settings to go across all of the computers, only the hotbars (I believe, Im really just typing up a faster version of the 'save files to something and manually move them between comps')