I see what Fairenuff is saying, and in healthy sibling relationships, yes, a husband would be secure enough in himself to be able to tell his sister that he is having a quiet Christmas at home with his wife and children. And generally I would never take over someone else's phone.
In this case, though, your husband doesn't know how to do this very simple thing. Give him a good, clear, easy example to follow, and he will see it really is okay to say, "No," simply, without any drama or emotions or hurt feelings or arguments or retribution.
From my own experience, I used to agonize over these types of dilemmas (and much worse) with my inlaws and wonder why my husband couldn't say a simple, "No, we've already planned to do x."
He didn't know how to, he'd never done it before. I think I was the first person to stand up to what was going on in his family. It felt GREAT. And you know what, there was nothing to be scared of and not a big deal. Remember, your husband and your children and how you spend Christmas are your territory, not your sil's or anybody else's. Internalize this truth, rejoice in it, and live it.
Go get his phone, right now, and text "Thanks for the invite but we're having Christmas at home. See you Boxing Day." Use his phone, and use it boldly. His sister needs to see that clear answer from his phone. If she thinks she can manipulate him by texting him and not bothering to ask what you had planned, then that thinking needs to get corrected. And now no need for nagging or worrying or agonizing. Don't get into any arguments with your husband. Let the sister do the nagging. Don't hide what you're doing. Go get the phone calmly and cheerfully and do what needs to be done. If he says anything, just smile and say to your husband, "We're having Christmas at home." That's all. Don't get into any long discussions. The fact is, it is not your sil's decision to make. Let us know how it goes.