That kind of acid will cause damage to you. If you start tearing up uncontrollably, then that is your eyes having decided they are taking damage from the acid and is trying to flush it off your lenses before you get irrevocably damaged and require a surgical replacement of some type for the eyes.
If it gets into your lungs then you will generate fluid to try and flush that way, the problem is you can drown in that fluid. Thats where hospitals come in with intubation and removal of fluids and so forth. If its really bad they stick you into a hyperbaric chamber to flood you with 100% oxygen while trying to eradicate the fluids. (Essentially a chemical burn inside the lungs at the aveola)
I would say forget the shop, stop at the first dealer or another shop with batteries similar to what you have on and ask if your company can be made to pay a immediate replacement bill for a new battery or batteries.
Frankly you don’t matter to the shop or company staff. Your injuries etc are your problem. Unless you can tie it to your company’s work.