If it boots without the battery connected, I would use it like that until you can get it serviced. If you can be certain this is not it’s normal shape, I would refrain from using it further with the battery connected, as bloated batterycells are always a bad sign and you risk fire.
I would suggest opening a case with the manufacturer first if its still under warranty, that way they should be able to get you a replacement even after you discard the faulty battery
Just send them the pic and say you discarded it for safety reasons. Maybe take another pic with something identifying like the original receipt or something unique to the purchase and a date for proof. Make sure to get a clearer pic of the bar codes on the top as well.
Knowingly mailing a swelling battery is a safety hazard and there should not be any issue with disposing of it as long as you have something to show it was the one in that particular laptop.