I have a 2015 MacBook Pro 13-inch Retina display. It seems as though the computer is bowing in the middle. Is this from the battery swelling?

If you mean the trackpad is being pushed up above the palmrest, or the laptop doesn't sit firmly on a flat surface, or you see the Bottom Case starting to separate along the edges, then it indicates the battery is swelling.

Regardless of cause, you can get a free over-the-counter evaluation (in warranty or out) at an Apple-owned store, FREE, provided you make an appointment ahead of time.

¿what is the date of your most recent backup, and by what method?

If you have to hand over your computer for service, the agreement you sign says YOU HAVE BACKUPS,

If the logic board is replaced, your same drive will NOT be returned to you.

My recollection (strictly from memory) is that battery replacement on that model runs about US$250, and replaces the entire top plate assembly, including keyboard and trackpad.