@bensleveritt
To me it looks like they're soldered onto the board and their solder tags are spot welded onto the battery case.
This is usual for batteries with solder tags because soldering leads directly onto batteries using a soldering iron requires too much heat and can cause problems in the battery so the tags are spot welded on in the factory under controlled conditions.
Unsolder the battery tag connections on the board to remove the complete battery unit i.e. battery with attached tags.
Hopefully the battery model number will be stamped on the other side of the battery. If so search online for the battery model number to hopefully find suppliers.
Also take note if there is a polarity marking on the other side as well i.e. + symbol. If so make sure that you connect the replacement batteries the same way.
You can damage rechargeable batteries if they're installed incorrectly i.e. reverse polarity