First, the bride and groom should contact the cousins and say "I'm confirming your RSVP the wedding that the two of you will be attending. Also, some relatives indicated to us that you may be bringing your kids and I am calling to just reiterate that the invite if only for the two of you, not the kids. You guys are clear on that, right?" and wait for them to agree. if they agree, then just quickly say "Great. I'm glad we cleared that up because we definitely do not have room for any additional guests and it would have been a HUGE problem if any uninvited extras showed up!" If they balk or try to guilt you then just hold firm and repeat "I'm sorry to hear that but this invite if only for the two of you. If you show up with the kids it WILL be a problem, that's why I'm calling. We don't want any problems for our guests or ourselves on our wedding day and I need to make sure this is cleared up now." If they continue to pressure you or try to guilt you then be firm and say "This sounds like a problem for you, one that we can't resolve. If you can't respect the terms of our invitation then we will have to revoke it and consider the two of you as a "no". I will let the family know you won't be coming after all." and then end the call.
If they show up with the kids, Someone needs to have a spine and tell them to make other arrangements or leave. If you are close with your cousin Rob, ask him to repeat the no kids rule to his houseguests and if they have the gall to try and bring them, have Rob text you in advance (you say they live a couple of hours away.) Then use those hours to contact Uncle Tommy and tell him to call his daughter ASAP and warn her to NOT bring the kids. In fact I'd call uncle Tommy NOW and tell him the family grapevine is buzzing that his daughter is planning to ignore your sister and bring extra guests to the wedding. Tell him she was told it wasn't possible to accomodate them, and it's going to be a huge problem if she ruins this wedding day by bringing uninvited guests and screwing up carefully laid plans. Ask him to run interference before the wedding and reiterate the invitation terms to his daughter, and ask him if he's willing to help the day of the wedding to ensure the children aren't put in an awkward position and that the bride does not have her day marred by party crashers either. If he's a nice and reasonable guy I think he can understand this and might help.
Talk to the reception hall and ask them if they have a staff person with a super strong spine who can handle turning away party crashers, and let them deal with it. No extra seats will be squeezed into the table and no extra plates will be made. And tell them they can't stay and that the couple was forwarned that extra guests can not be accomodated for and must leave.