Battery busses are much worse than trolley busses or rail. They're also significantly more expensive than all other options.

Their only upside is nimbys might let you have a transit system if you suggest them.

Are there any studies to back that claim the EV busses make public transport safer?

Yes, it's mostly air pollution from the engine and braking that is reduced:

https://www.transportenvironment.org/wp-content/uploads/2021...

Oh, so the busses are not safer in terms of collisions or pedestrian fatalities.

How do you account for dangers of battery production and disposal? Are injured EMTs and mechanics, due to high voltages, also considered? How do you account for pollution of installing upgraded transformers and lines to the bus depot for charging batteries?

To be fair, a sanely designed bus system (ie. RBT or busses that aren't useless because they're stuck behind a car) is just is safe as light rail...but then you can just build light rail because it's cheaper.

I usually expect the people doing Lifecycle Cost Analysis to think of these things and quantify them.

Have they all missed these deaths that you've uncovered? Enough death and destruction to overwhelm the well studied impacts of air pollution?