You can dissolve sodium bicaronate in water (use a baking soda with a minimum or no additives) and really soak the wood through. I mean for several days, if possible. If the areas isn't watertight to just pour in the neutralizing solution, at least use some soaked clothes to give the neutralizer time to penetrate (after all, the acid had time as well...)
Using too much sodium bicarbonate solution is no problem, leaving acidity in the wood is! You can also buy some litmus strips to see if any acidity is left after treatment.
Plastics marked wtih FEP, PFA or PTFE are resistant to sulfuric acid, although you can order sheets of them to mould your own, I would look around Home Depot for something moulded from PTFE that fits your requirements. The thick black plastic sheets used in gardening are made of HDLPE, which is resistant to sulfuric acid. You could seal your battery area with an
epoxy and cover that with a layer or three of sheeting.