But why would/should he see it as his problem?
He's invited you to a bbq, if you can't make it it's not his fault. The reason isn't really relevant. It's no more his problem if you can't make it due to not having childcare than if you couldn't make it due to feeling ill or having something else booked or your car being in the garage.
You're making it into a huge drama, when it's just a random weekend BBQ! He invited you, you choose whether to come or not. It's as easy as that. He doesn't have to bend over backwards to make it as easy as possible for you to attend, you don't have to attend if it isn't convenient, for whatever reason.
You both seem a bit odd, with terrible communication issues.
Yes, you're right in that most people probably would invite the whole family to a bbq. But it's not particularly odd that someone who doesn't have kids doesn't invite everyone's kids to every event they host, nor is there anything wrong with that. Personally if the invite specifically said 'Do you and Tom want to come to a BBQ' rather than 'do you and the family' or 'do you all want to come,' or 'bring the kids,' then I would have checked asking, and if the answer was no wouldn't have been offended. Equally if I hosted an event that kids weren't invited to, and some people couldn't attend because they didn't have/want to get childcare, I wouldn't be offended.
As per the MN saying, it's an invite not a summons!