No, he doesn't do that with the bottle. This is called nipple confusion. Babies this age will pull from the breast when the flow of milk is slow, especially if they are used to bottles. Watch the video clips of babies drinking well at the breast, or not (see them at NBCI.ca), so you will be able to identify when your baby is drinking well or not. Observe your baby at the breast: Chances are he starts pulling when the flow of milk slows down.
So what can you do now? Well, you can increase the flow from the breast. You can get a better latch (see NBCI.ca). You can watch for how well baby is drinking. When baby is not drinking very much, before he gets upset, start using compressions (see the information sheet Breast Compression at the website). If that stops working, switch sides and repeat. Domperidone, a medication used for increasing milk production can also help. It can be obtained in the USA, even if you are told it cannot be.
If the baby needs supplementation, then using a lactation aid at the breast will keep the flow up.