Samphire has this spot on OP.
The most concerning part - apart from you needing to ask if YOU are the unreasonable one - is this:
I've asked him to come to bed earlier or sleep upstairs but I know he's going to be defensive and say that it was the dog jumping on me, not him who woke me up.
You have not spoken up because you have already anticipated that his response will be to dismiss, minimise, & lay blame elsewhere.
He's moved into your home, & doesn't even have the courtesy to treat it like mutually shared territory. He thinks nothing of waking you, because he cares more about his own selfishness than your sleep patterns. He feels his entitlement to gaming & 'inevitably' waking you up is more important than your sleep, health & happiness.
He must know you are struggling - anybody decent would OFFER to sleep on the couch/spare room if you have one. But he insists on waking you. And refuses to be accountable for it.
What other ways does he demonstrate this selfishness to you OP?
Because - forgive me - here you are - an intelligent, capable, resourceful A&E doc, reduced to wondering if it's ok to challenge her fiance about his habits. So I can't help but wonder how long he has been dismissive, minimising, & unaccountable around you, & what part that has played in grinding you down to the point where you are wary of challenging him.
It's exhausting to constantly have to struggle to get heard. To be treated decently. When you reach the point that you have to keep explaining to a partner what basic levels of kindness & consideration are - you are flogging a dead horse. Couple that with the PTSD-levels of constant A&E stresses staff are handling, top up with sleep deprivation & ... well - you're the doc. You know how perilously close humans skate to the line between coping & breakdown.
So you need to put yourself first. You can't help anyone until you have helped yourself. Do you feel able to challenge his defensiveness strongly enough to finally get heard? Or would that only end up in a blazing row, more blame, maybe some stonewalling or a nice dose of DARVO? That's how selfish people tend to respond to being challenged. So don't do it until you've managed to catch up on some sleep & do some thinking - in fact, don't do it at all if you don't want to! - but at least his response would give you information. You could then use that information to decide if it's worth your while continuing cohabitation with him.
Please ensure your contraception is watertight. Apologies for how patronising that sounds. But you can imagine how he'd be around ensuring equal shares of sleep, with a baby in the mix, yeah?