Backing up to an existing Time Machine sparsebundle
As it turns out, I needed to make the user whose account is being used to authenticate to the Time Machine mount point from the remote laptop the owner of the MightyMac.sparsebundle file. I discovered this by comparing the permissions on the new sparsebundle it created with the old one I wanted to backup to and found the new one was owned by the remote user and the old one was not.

After I made the remote user owner of the old sparsebundle Time Machine backed-up to the old sparsebundle file.