I was not able to find the exact match by size and capacity

Any 3.7V Li-Ion/Li-Pol battery you can fit will work.

You can use a few smaller batteries like the original did.

You can use a smaller battery, or a bigger battery like I did

Use a replacement battery only if it has a protection board. I have no idea if a puck PCB has any kind of protection circuit built in. Better safe than sorry.

I ended up using a rechargeable 3.7V battery from a disposable vape (Elfbar 800).

This battery did not fit the puck case without modification, thus, I had to remove some plastics that held the original battery in place