It really depends if she’s being 100% honest. Or if it’s an excuse. Only you can know that, OP, from your relationship with her
I’d assume she’s being honest. If she’s limited to only 6 kids, she obviously included your DS in that small group. Once mine were at school, if numbers were limited my DC wanted school friends. So I couldn’t always invite pre-school friends (such as kids of my mates). She has included your DS in the small number originally invited.
I can understand her reasoning if one child has to be uninvited. If the 5 invited kids are all from her DS’s class, it’d be awful having to choose one to retract the invite from, who will have to hear them all talking about it at school, likely feel sad and wonder why it’s them. Whereas your DS at least wouldn’t have this. Plus, they probably all know each other, whereas your DS will only know the birthday boy, so may be feel a bit shy with the other kids and cling a bit to birthday boy.
What I can’t understand is why she can’t pay for another place on the activity. That’s what I’d do if I’d made this mistake. Or is it something they can only do as a 6 or costs a huge amount per child? If the latter, could you offer to pay for an extra place for your DS?
Could a get together separately, where your DS gets birthday boy all to himself, actually be more fun for him, rather than being with kids he’s never met?