What's a web of cuckoo? I've never heard this phrase.
It's a bit complex, but there are numerous ways to deal with people, particularly if we are attempting to change behaviours. It's particularly challenging when someone is in complete denial, as this lady is. The OP thinks Dobby wants to be with her and that she has the better home as she is around all the time and Dobby gets lonely on his own.
She is ignoring and rebutting attempts based on ownership. An alternative strategy needs to be used, one which puts at the heart of this matter, not the ownership of Dobby, but his safety and well-being, based on issues other than whether he's lonely. In certain instances, it's necessary to go in hard. By focussing on his health, the OP is reinforcing the fact that it is she and her husband who know Dobby's health situation, not the old lady and this further reinforces the fact that they have this knowledge because he is their cat. Including a vet in this scenario is also important. Vets are professionals, they are authority figures and most people recognise them as 'truthtellers'. If a vet says Dobby needs to be kept in one home, he should be. Also, it is now practically impossible for her to claim she looks after Dobby better than the 'neglectful' OP, because he having ongoing treatment with a vet.
Yes, it's a lie, but it's more than that. It's a manipulation of a manipulator. The old lady has lied repeatedly, discussion over a period of time has been unsuccessful. This situation needs shutting down and for that, they need to present a faits accomplis.
As I said, I know psychology can be quite complex, but I hope I've explained it as simply as possible. When discussion is unsuccessful, you need to create a dead end. Something along the lines of a serious health issue would work, because the old lady can't argue against it.
I can't remember whether Dobby's microchipped? He needs to be and I would probably contact all the local vets and explain the problem you are having with this lady trying to claim ownership of your cat and ask them to check the chip of any cat matching Dobby's description prior to treatment.