I don't think the rehoming was likely to be as sinister as is discussed on here, or implied by their silence. As a settled couple owning a home, a dog likely made sense to them and fit with their lifestyle. Once they split, and once Martha stayed with Tanya, it may well be that her lifestyle didn't allow for Martha to get the exercise and attention she needed to be happy and healthy, and the decision was made to rehome her where she could be better cared for.
People might judge others for rehoming an animal, but I will always prefer someone doing right by their pet than choosing to leave the animal unhappy to avoid judgement or doing something hard.
The silence on the matter, combined with Sarah finding yet another excuse to take a jab, is what's made it such a scandal, I think.
I do wonder if Jim was offered Martha, but preferred her to be where he could visit, maintain contact, without impacting his new lifestyle and relationship. Hell, he might not have been allowed a pet where he was.
If it was a case of Tanya wanting to rehome Martha and Jim not wanting that to happen but also not wanting to take her himself, I could see that causing a lot of fighting and bitterness and an agreement not to speak about it publicly.