I've rebuilt a few twin hammers. You're going to want to use a thin coating of a moly EP grease. But a lithium complex that's EP rated is also just fine. You're NOT going to want to inject grease into the hammer chamber. It's a waste of time and usually hinders performance. I know there's a grease port, but it's useless. If you want to grease a pneumatic impact, you need to split the case, clean the hammer with a solvent and manually apply fresh grease.
Most pneumatic twin hammer guns have two sealed chambers. The hammer/anvil/nosecone chamber and the pump body. The two don't mix. You need to keep up with the grease in the nosecone and use air tool oil for the bearing and vanes in the pump body.
You should use air tool oil at every use. Put a rag over the exhaust outlet and blow out the excess by operating the tool for a second or two.