I understand .. truly do. And it's an odd thing but sometimes older dogs actually prefer being OUTside than in when not feeling well. I always thot it was sort of a 'call of the wild' thing but my holistic vet said simply it is often because the heart isn't functioning at full capacity and being outside literally helps them feel like they are getting more 'air' - more oxygen.
She has access to 'shelter' and won't be lying on cold concrete unless she WANTS to. She probably felt as bad about the mess as you did ... and she indicated HER choice by telling you she didn't want to come in.
Frankly, you sound to me like an incredibly loving, but worried, dog mom who did the best that she could in giving her dog the best options at hand while ensuring safety as much as possible.
I kinda doubt if Dakota is feeling puny, that either one of them are going to jump the fence to visit the world. And I'd bet that the other dog will stay to keep Dakota company "Just because".
You can give dogs Kaopectate ... but it sounds to me like maybe she needs some slippery elm, but the vet will do a fecal and will likely give you whatever is needed.
It's a difficult time of year for the elderly, truly is. I'd imagine part of her worry was the fact that she WAS making a mess for you. On a lot of levels I'm sure she appreciates what you've done.