I updated my local hosts file and the WordPress install worked on either computer seemingly fine.
I made sure that AMPPS and therefore MySQL was only on a single machine at any time and made sure the Dropbox itself was always up to date.
It looks great, but I get database inconsistencies all the time rendering the installation useless. Looking in the Dropbox/AMPPS/var folder where the database lives I see the attached.
Sorry I am not able to get it as we have never tried using Ampps with Dropbox.
But I would suggest you to create a symlinks pointing towards dropbox folder for specific folder in Ampps.