What size PIN Battery

@christolomia The key to these lithium ion batteries is that six digit number, in this case 803040. Those are actually the dimensions of the battery; this one is 8.0 mm thick, 30 mm wide and 40 mm long. Searching with the term "803040 lithium ion battery" will get you several sources for that particular battery; the main thing you'll need to pay attention to is the kind of connector it comes with. Here's one that sounds like it may work.

Amazon.com: AKZYTUE 3.7V 1000mAh 803040 Lipo Battery Rechargeable Lithium Polymer ion Battery with JST Connector : Health & Household

You may find other batteries that are compatible except for the connector; if that's the case you would need to change out the connector that comes with the battery for the one on your existing cell.

Note that you can always use a smaller size battery; of course that will reduce the mAh rating so you won't get as much life out of each charge, but otherwise it will work fine. I assume you wouldn't be able to use a larger battery due to physical limitations on the space the battery occupies.

Hopefully that'll get you going; let us know how it all works out for you!