But what will a coin cell do when it's trying to get charged? That sounds a bit worrying.
But I didn't realise that. Perhaps I could use a voltage divider then or simply input 5V. 5V is too high for a battery but most devices accept it there anyway.
> But what will a coin cell do when it's trying to get charged?
You can get rechargeable lithium coin cells. It will just recharge fast.
The cell usually has enough internal resistance and surface area that it can't overheat or violate the charge current limits.
I still wouldn't recommend this approach, simply because the DC/DC converter approach you specified is slightly more power efficient (onboard chargers tend to have poor energy efficiency - many are simple linear regulators from 5v!).
I use cheap ones from Amazon (link below). I power them with 12V adapters which I "shucked". 5V is not enough to provide enough voltage because you lose some in the conversion.
These adapters seem similar to what's used in that instructibles link. I didn't pick these for any specific reason other than they were available on Amazon. It uses the same LM2596.
You might want to check if the battery is going "spicy pillow".