You say as soon as you put baby down eyes are wide open. And then what? Do they scream or.. ?
Is baby content when awake? Maybe try a baby gym or something now they will start grabbing at things so just pop them down awake in a safe place and you should be able to have a few moments with your partner! It gets better as they get older
Have you tried incorporating white noise during feeding etc and not just when trying to put baby down? That way, he might associate the white noise with calm/relaxation?