DS2 is really clingy, I'm sure, partly because of his personality, partly because of the attention demanded by his older siblings. He's hardly any less clingy of DH (who hasn't secretly been breastfeeding DS2...well, not that I know of, anyway).
Your doctor is completely dismissing the good feelings that breastfeeding brings about (see, ignorant MAN again).
Do you think it might help if you could write up a schedule for things like bathtime, meal planning?
And could you bribe/incentivise your kids to tidy up for you -- say, no pudding until they've had a whizz around the place to tidy up? What if, on non-bath nights (say, 2/3 nights/week) the 3 older ones earnt a pasta piece (for the jar) for entertaining the baby for 1/2 hour while you get some housework done. This would be on top of the "no pudding until your junk is all tidied away". If someone refuses to tidy up then the others get pudding and an extra piece of pasta.