It's a horribly distracted age. How is his weight gain, wet nappies etc? Sleeping like that would suggest that he really isn't starving. It could well be that he's just frustrated at having to work for milk he's not all that hungry to drink, hence the bottles going well 'cause he doesn't have to work for them. I'd be wary of giving him too many bottles as you could find that he starts to refuse the breast all together. Either that or using something like a medela special needs feeder (ex- habermann feeder) which mimics the work needed to bf.
You could try some breast compressions (see faq) to speed up the flow, or let him switch sides back and forth. It's kind of irritating for you, but it does mean he gets quicker access to your milk. Feeding while sleepy is another good trick - we get our best feeds in a darkened room just before nap time, or right on waking (if I've woken her) - too awake and she has better things to do than drink.
