I would respectfully decline, and send a card or something.
I got married this past Christmas Eve, it was a tiny family-only ceremony and of course kids were invited, our son was 11 months old at the time and, in fact, kids probably made up 1/3 of the guest list. It would have been awesome, the kids were playing together afterwards and having a ball.
As I was downstairs getting ready, 5 minutes before the ceremony, the clerk came down and mentioned that one of the guests had brought a little boy who was quite ill. He was, apparently, vomiting everywhere, whiney, and begging to go to bed.
Turns out, my SIL's little guy (2 at the time) had a raging gastro bug. And since her husband was sick too, and their 7 month old baby, she decided to bring the kids with her to our freaking wedding
so her husband could sleep
This was a tiny wedding too, like 25 people/kids, so we were all at one huge table for dinner. He vomited all through the reception, people had to jump out of their chairs to dodge it at one point.
Long story short, everyone got sick. My brother and his wife ended up with the bug not 4 hours after the wedding was over. They had come all the way from the Bahamas for a family Christmas in snowy Canada, and couldn't even get out of bed to open their gifts the next day because they were so sick. It lasted the rest of their visit, it seriously sucked
Looking back, though, if I could do it all over again I wouldn't not invite kids. I'd specify on the invitations that common sense was mandatory, though! But honestly, whereas before I might have been a bit grumpy at a 'no kids' rule, I can see it from both perspectives now. And I don't think I'd go to an event my children weren't invited too, but that's just me.
Hopefully you can find a way to work it out! Good luck!