Those don’t appear to have the protection circuit board installed.
You can buy lithium polymer cells with circuit board already installed on eBay. Some eBay sellers will install the connector if you provide details/photos.
I measured the compartment to take up to 43mm x 72mm pack. Max thickness 4.0mm. I found cells on eBay that would fit with a capacity up to 1200mAh.
For the DSO connector, you need a Micro JST 1.25mm 2-pin connector. I couldn’t find a suitable crimper tool, so best to have the supplier install the connector or buy pre-wired connectors.
I purchased a 404060 3.7V Lipo ($10.99USD with shipping) and pre-wired micro JST connectors (20pcs $2.39USD with shipping), both sourced on eBay from sellers in China. Lipo was shipped by Sweden Post.
I carefully removed and reused the kapton tape. Make sure the red/black wires are correctly polarized in the connector before soldering the wires to the battery protection circuit. I had to reverse the wire positions on my pre-wired connector.
