Im torn in this one.
I can see why your DH got so cross about it, but more that his own DM organised/suggested it to you without first checking it was suitable for her own DSs dietary needs, which suggests shes not quite beliebing/supportive of his situation - so to me, MIL is BVUR & that would have no doubt stung & added to his over reaction.
YABU for not understanding that, though YANBU for wanting to go & enjoy this family meal & wish him to come to.
That said, going solely for a meal for a birthday treat, is not the same as going for an activity as a birthday treat & eating whilst there as you did for DHs birthday, so YABU there too.
DHs is BU for kicking off so much, but I do think you need to cut him some slack & double check the restaurant staff got it right, as I agree it sounds off that they can't cater for 2 of the most common allergies & I think them offering that you bring a takeaway is pretty crap TBH.
If it were me, I would ask the restaurant to cater properly for DH, or I would take my business elswhere, so based on that, as much as I do understand your POV, I think YABU