Check the power requirements on the projector as well as the power output of the transformer just to make sure the transformer can support it. I don't see it being a problem but a test run would be good. The only issue I could see would be blowing the fuse on the lighter. That won't hurt anything though, that is why the fuse is there.
One of the inverters I have (I believe I got it at Walmart) has the ability to go directly to the battery. Getting a second battery its a great idea though.
Purchase a Honda EU 2000 from Mayberry sales. You will be happy you did.