It sounds like you're putting a lot of time and effort into helping your boyfriend, I think that's really commendable, because it can be really difficult.
That's great news that he's seen his GP, and that he seems to be having some good days. I've always found that with depression, on the good days you're yourself, and on the bad days, you're a different, sad, inconsolable person; it's not just a sad you, it's like being an entirely different person. It might be helpful to remember that when he is feeling down, and saying things which might upset you, that his brain isn't doing him any favours, and it's just making all of these horrible chemicals that he has to ride out, and so sometimes those chemicals make us say things that really we don't mean or even feel underneath.
It might be risky to talk to his doctor, as it could make your boyfriend feel a bit alienated, as if you and the doctor are working together, and he's not included. I know that sounds a bit silly, but when people are feeling severely depressed, it can be difficult to really grasp what it's like if you've not felt it yourself.
I've always found help with counselling, and I know lots of people who suffer from depression who find it comforting, calming, and productive. It could be good, though, to bring it up when he is having a good day, so he can better feel a part of the decision and understand why you're suggesting it. If he's down, he could feel as though you're palming him off to a counsellor so you don't have to listen, or that you think he has real 'problems'. Again, I don't mean to sound dramatic, but being cautious can make sure you stay on his side, which is necessary to progress through this together. Perhaps when he's feeling good, you can ask if he would be interested in it, because it would give him a chance to talk freely with no judgment, and that he can be open and honest and cared for by the counsellor, and not have to worry about what he's telling you. I always loved coming home from the psychologist feeling like I've had a rant for an hour, but my wife didn't have to hear it!
The most important thing, though, is to make sure he feels you're on his side. It's a super difficult time for you, and you may be tired and hurt, and no one is taking care of you (except us here on the forum:) but try to stay as a unit as much as possible, and make sure he knows you only want him to be better so you can be happier together.
Hang in there, you're doing great,