dead balls are actually easy to play, but the main reason for missing that, is that - well they're dead - so the timing is different, and you have to adapt your footwork. Rushing - or taking the ball too late when its low and you're standing a bit too far from the table, is going to cost you the point.
I think the main focus should be to be moving well and get in the good position quickly and then everything else is easier. I think taking the ball a bit earlier helps with a close bat angle and don't put too much power in the ball, make sure you feel your body is going a bit forward at the end of the stroke. Not just a vertical stroke else its a miss.
The other way if you're late on the ball is just to lift the ball from under and put it back calmly and gently on the table. The kind of player you seem to be describing will not be an all-out attacker. so just play that one more ball and wait for another opportunity.
the guy with anti cannot attack you with topspin. His best chance to score a point is your mistake, or to do a flat hit if the ball is high. so if you just put back the ball on the table , without giving a high ball, while moving him a bit, he's probably not going to hurt you with his anti. Unless his BH is fantastic and he can attack with it you should be ok.
Because the anti player has few attacking options, if you can stay in the rally, then you can afford to be patient and wait for the inevitable easy ball coming to kill .