CAn a jump starter power an trailer slide out?

Yes you are right. He did say he had a brake controller. So as far as I'm concerned operating the slide without a house battery, is far down the line on things to worry about as opposed to operational safety equipment!

I hope the OP considers all this. It's not like the trailer is parked permanently at a campground, he said that he tows it, so all safety systems need to function properly.