I agree with this.
I think that the birthday v stag do has highlighted the long-term difficulties you are both coping with.
8 week old first baby - which is a huge adjustment
struggle to make sure you can get enough maternity leave
overtime and no time for family time.
Sole responsibility for baby due to heavy work commitments
Little available support.
On one hand he's agreed to be best man and its harder for him to bail on Best Man commitments. He's also probably looking forward to his first break from work in a long time. Perhaps he didn't realise how many there were. You both sound quite young. Perhaps this is the first one he's been to and its a long term friend etc so he feels obliged. But it does sound as though he many not have fully thought thru the cost and time implications. And he needs to do that and work out if he needs to cut back a bit on the trip.
On the other hand if your only planned break (and a lie in and bit of time off from baby is not asking for the moon) was not scheduled to happen until May and is now scuppered I can completely understand why you feel so disappointed.
Things will get better, having your first tiny baby takes a lot of getting used to and you are still very early days.
First off make a new birthday plan - go stay with your mum for the weekend and have a family birthday.
Second off plan a family birthday weekend, where you have breakfast in bed, a lie in and a family day out with baby. In my experience - EVERYTHING seems better in the open air, out of the house... a nice park/cafe / stately home grounds sort of thing (if that doesn't sound too boring) and then maybe a takeaway and movie at home.
Third. You need to find some practical solutions to getting a break from being on baby watch every second. See if your mum can commit to a regular day when you visit her or she visits you - maybe on a day when BF is working. If its regular, whether once a fortnight or once a month, it will be easier for everyone to plan around it. It also helps you to know that there's some help coming up.
When you are more on top of baby's routine, Join some baby groups where you can make baby friends (lol) its a good way to get out of the house and have a break that way. Plan some nice regular walks - we used to go to a garden centre which had a big aquarium - it was all free, calming walk, cafe nearby and the DC loved staring at the fish/plants. Even going to the mall for window shopping was a great distraction. Find some easy local things like that. It will break things up and take some of the pressure off.
Talk to BF and see if he can manage a regular day off - 7 days a week is not sustainable. Also its important for him to have enough time with the baby.
Make sure you are taking good care of your self. Vitamins, healthy food, (nap when baby naps was great advice for me if you've been up all night) don't worry too much about housework etc... that will pick up. Building a good routine helps (that one took me a while to work out)
Everything with young children goes in phases. Sometimes you think I can't cope with this all the time, but remember its phases, they will be onto the next phase and it will be a different challenge but often it's an easier one. so just take it all one day at a time and you'll get through this.