Is he indicating that he is finished after 10 mins of nursing? Sometimes babies get bored when the flow slows down before the next let down, so perhaps he is stopping too soon. Also giving a bottle after nursing can make him expect that and then he in turn gives up on nursing sooner, which causes lower supply and then the whole thing gets exacerbated. And even a full baby will continue to take BM or formula from a bottle after a feed so just because he takes 4 oz, doesn't mean he needs 4 oz. Is he having a good nurse when he wakes during the night or just comfort
There isn't breastmilk that's not "rich enough" as someone mentioned. But if the baby isn't fully draining the breast he may be missing out on the fatty hind milk. The fatty milk sticks to the ducts so if he's not nursing long enough he will be getting the carb-rich foremilk but missing out on the fatty stuff and then he'll be hungry sooner.
If you up the formula your supply will go down, which is fine if you are planning to transition away from the breast, but if you want to continue to breastfeed there are things you can do to protect and build supply. You can call a local La Leche League leader for some suggestions.
How is his weight gain? If it's not good, the doc may be right that he is hungry but if he is gaining well, that may not be it. Also 4 mos is a growth spurt time so it just may be an extra long one and he really does need all the extra feeds. I know my DD went through different stages where she'd suddenly be nursing like a newborn again (and driving me insane!). So I nursed more, the supply picked up and when the growth spurt was done she went back to normal.
Have you considered cosleeping? There are ways to do it safely. I think Dr Sears or Dr Jay Gordon have info on their websites. With a baby who nurses all night, it can really be a lifesaver.
It's not time for cereal. It does not help babies sleep better and can be counter productive if it doesn't agree with the baby. Sometimes babies who want to eat all all the time have reflux. They cry because they are uncomfortable and want to drink because it feels better while they are drinking. This can be exacerbated by large meals (just as heartburn is for adults), so frequent smaller nursing sessions can work better.
Sorry, I don't have an easy answer for you and I just threw out a bunch of different possibilities. I know it is so difficult when the baby isn't sleeping well. I hope your sort it out soon.