What's the difference between fight and battle when they are used as a verb?

Two points.

Firstly no matter how big the battles were in which your grandfather took part his individual contribution would have been on a personal scale.

Secondly the description of doctors 'battling' to save a child's life is a metaphor. They are not fighting like soldiers they are doing their very best, medically, to save the child. They aren't fighting against other people, they are working against the infection or other physical condition that threatens the child.

The words 'battle' and 'fight' are synonymous and equally interchangeable in your example sentences. Battle is derived from the Old French word 'bataille' which means to fight, whereas 'fight' has Proto-Germanic origins from 'fechten' which means struggle with.

References:

https://www.etymonline.com/search?q=battle

https://en.wiktionary.org/wiki/fight

Verbs:

“To battle” is pretty rare. It’s meaning is almost synonymous with fight (two dictionaries used fight in the definition, e.g. “to fight or struggle against”). It always seems to have a more dynamic and ongoing aspect - unrelenting and continuous, if not intense.

Nouns:

In military terms, a battle is usually seen as a single incident of engagement at a location. May be long, but is one engagement. There may be exceptions in a military context, but those are from the non-military usage creeping into a military context (“The battle for control of the Pacific” may be military and could be used now, but is an example of what I’m saying.) I feel pretty confident that historically, a battle was always this one engagement at a location.

Even in non-military usage, battles have the feel of being, if not an engagement at a location, none the less a potential subset of a conflict between the parties. There is some notion of battles having a domain. You’re more likely to hear about a battle for something or a battle over something or at somewhere than you are to hear about the battle between parties generically. As if there is a theater whether it be topic or location or object of control (the battle for young minds)

A fight is much more general. Fights can sometimes just be between parties; battles are about things. Feuds are fights but are not battles. Fights can go on through time and space, but they don’t have to. That’s part of fight being a more general term. Battles are open, active conflict within a domain. Perhaps that’s why I mentioned ongoing, continual, and intense with respect to the verbs.

Even if I fry to get away from that aspect of a battle with context, as in

The ongoing battle between Bob and Bill.

My mind at least is expecting “specifically the battle over” such n such.

If I had more time this would be shorter. Hopefully that helps some anyway. Great question. Gotta go.