OP doesn't seem to be accepting that she may have upset her SIL, and there's a lot of posters saying that they can see how she felt hurt.
Op, in fairness, if you'd said that once everyone else had gone home, or if in a different scenario where you hadn't spent do much of your time and energy helping your SIL, she may've laughed too, and just thought you were being daft.
However, although you made sure she knew you'd had a good day with her at the time, if she was so thankful to you for all that you'd done, and the hours you devoted, she may have felt a little insecure, and then when you all got to where the dog was waiting to greet you, what you said, and tge enthusiasm with which you said it.
It really sounded natural and spontaneous.
Your earlier politeness to your SIL, could of then seemed like a duty remark and not genuine like your response to the dog.
Just tell her "omg, yes I can now see how crass and thoughtless that sounded. I was just as daft with my equally daft dog.
I would wish to hurt you at all, so please accept my sincere apologies for my unintended eff up".
I've unintentionally upset people in the past, and its horrible, coz I really hadn't engaged my brain enough to see how someone could come across.
If you stick to your guns of, that you didn't mean the way she took it, will that be your consolation wgen next your due to be with her, and you know that's she's sad... doesn't matter if you or other posters thin she's being over sensitive.
Other peoples levels of sensitivity will always likely to differ- who's to say who's right or wrong?