I'm not familiar with the add-on you mention. If it writes to the usual contact category field in Thunderbird local database and doesn't have its own way of storing contacts or contact category, then the following may apply:
Usually if you want to synchronize Thunderbird categories to Android, it is necessary to select the "account" virtual address book mode. That means that each account on Android will represent an address book. This way, a contact belonging to a specific account/address book on Android can belong to several contact groups. By contrast, if you select the "group" virtual address book mode, it means that each group on Android will represent an address book. In this case, a contact belonging to a specific group/address book on Android can belong to only one unique contact group.
To select the "account" virtual address book mode:
- In BirdieSync main window on your PC, select Configuration view
- Select your Android source in "Sources" column
- Select Contacts in "Contents" column
- Above the address book list, click on the gear button on the left to open the "Address book" configuration window
- Select "Each contact account represents an address book"
- Press OK to save the configuration
Now you should have one address book per Android account, that you can synchronize with Thunderbird.
If you want to synchronize a unique account on Android, then you can only synchronize it with a unique address book in Thunderbird.
Note also that instead of using Thunderbird categories, you may prefer to create different address books in Thunderbird and synchronize them with Android contact groups, using the "group" virtual address book mode.
As usual before making any change, it is recommended to backup your data as mentioned in this faq entry