I'd leave the kids and go :) But that's just me, and after two years with twins I bet Ill be dying for an adult only affair. Sounds like so.much.fun. Anyway, I agree with other posters, just check "not attending". I'd make a donation, and if asked I'd say that Id love to attend an adult only wedding and it sounds like a blast, but it just wasnt possible for me at this time.
How to politely decline an invitation to an adults only wedding? - Page 2
If you don't want to entertain working up to them being left with someone else, I'd travel all four of you, get a hotel near the venue, and then have your hub stay with the kids and you go to your cousin's wedding.
FWIW, I didn't leave my first born for more than a couple hours until I went into the hospital to have my second born (he was 2-1/2), and I was gone for 3 days. He wasn't nursing at the time, but still. BIG adjustment. DH brought him to visit, and stayed with him most of the time (my parents handled the actual birth time and when DH was with me initially), and he did FINE.
I left my DD for the first time to go to a wedding of a special friend when she was 18 months old and still nursing about 5 times a day....and I even stayed overnight...and she was fine, too. A little sad here or there, but my parents managed (though my mom did call me at 8am to say she had been up since 3 and was there any way I could get back sooner rather than later LOL).
Anyway, I think you should go to the wedding even if you go by yourself, because it sounds like you want to go (and I don't blame you, it sounds very special!!) Going to functions by yourself can be a lot of fun.
I reread your OP and saw where the nearest hotel is 2 hrs away, but if you really, really want to go, but don't want to leave the kids there might be a way that you could go for a shorter time and have the kids nearby.
Some former older neighbors across the street used to have his son and the son's family come visit. Their house was small, so rather than stay with Grandad and Grandma the son and his family rented a small RV and parked it in the driveway and used that as home base. I don't know about the area where the wedding is, but it sounds like there might be some forest land or something nearby where the kids could run and play with dad or a babysitter while you go to the wedding and maybe an hour of the reception. It would be expensive, for sure, but if you really want to go that would be an option.
Otherwise I would just send some really sincere regrets and not get into the whys.
But yeah, it also sounds like you want to go. If that's the case, try to work it out! Maybe RSVP with an explanation that you may need to change your plans at the last minute in case things don't work out. That's when it might make sense to explain that parents of young children simply can't plan for future events with as much certainty as child-free people. Clearly they will understand that, since I'm sure it's one reason they want to be child-free.