H.W.
First, you mentioned that he ate a lot with the bottle, so to me that suggests it's easier for him to get the milk out. So, if it were me, I'd pump and use that milk to feed him.
One thing which will happen in daycare is that your son will likely start to get on some sort of feeding schedule instead of 'demand' feeding. I've worked as a nanny and at daycare; it's impossible to feed 8 infants with three caregivers all on 'demand'... we just don't have that many arms! :) Daycare workers are also going to let your little one fuss a bit, not out of neglect, but because A. we aren't the bio mom and don't have that physical pull to sate a fussy baby right away (it's true, as I've experienced this as a mom and as a caregiver) and B. we are constantly having to prioritize which most necessary task comes first.
What will happen, though, is that a good daycare will document those feedings, when they happen, how much was consumed, BM/diaper changes, etc, and you will likely start to see a pattern developing. I encourage parents to begin to adapt their own at-home routines to what the child is experiencing at daycare, so that on the weekends/days off, they are fed consistently. This will also help with your son getting more food.
if it turns out that your milk isn't producing enough fats and carbs, then supplementing with some formula would be ideal. If there's an issue with how he is digesting/processing the milk, that's another category. But you do want to move away from the 'snacking' feeding and toward fewer, larger feedings. (when my son was wee tiny, he'd nod off every time he nursed. We had to take off his socks and tickle his feet to keep him awake long enough! So I understand your concern. Hang in there!)