It goes without saying, I don't know you , your elderly relative, her dog, the situation or the history. However you've provided some information and are asking for help from Mumsnetters in deciding how to do the right thing. So I will offer response, based on the little information I have, and how it seems to me, with all of the above caveats.
It seems to me that your elderly relative could use some really good support from people who truly have her best interests at heart, will truly listen to how she is, and what - if anything - she could use some help with, someone who will offer support on her terms.
NHS website says dementia affects people differently, progresses differently, and with the right support people with dementia can often continue to live their normal lives independently for many years. Many people can do the same with poor mental health and personality disorders. None of these conditions automatically preclude people from taking good care of a dog.
You say your relative is forgetting to feed the dog. You also say you have t seen the dog, or your relative, that your relative no longer likes you, won't discuss situation with you and won't llet you in. So it seems you may not have first hand knowledge of how dog is treated. There's also a sliding scale of 'forgetting to feed the dog' from the normal hustle and bustle of family life where adult teen forgets one lunchtime, to a dog who used to be fed 3 meals a day and now gets those meals at different times or now gets 2 larger meals etc right up to missing meals in a way that affects dogs weight. You've not given enough information to to say what's going on. However it seems that a reliable system of reminders and perhaps repeat deliveries of dog food and regular drop ins could help solve that completely.
The facts seem to be that your relative has been recently assessed by professionals as having capacity to make her own decisions. Social services and her GP have been involved. You dont seem to be directly involved but can raise your concerns through these channels.
It seems to me like your relative needs true friends who will be there for her and support her to live well as she wishes and within her capabilities. I hope if I ever find myself in that situation I'll have that, and if I can't have that that I can rely on social care and health services to support me within law, respecting my wishes and capabilities and offering me help if I need it - true help - in applying for funding, getting dog walkers, support at home if wanted, equipment I need for reminders etc.
I would say, since you asked, try to be there for her and her household, which includes her dog. If you cant for whatever reason - be that due to old animosities, your unwillingness, or hers, simply try to ensure she continues to recieve whatever support she needs from health and social care services. There is a lot of great support available for her to access for this situation, to support her remaining happily and well at home, living her way.
The dog's wellbeing matters a great deal of course. It sounds like your relative may be doing her best and may need support to do better, the way she used to. The dog is her family - who knows, perhaps her only true friend -ensuring your relatives wellbeing may be the best way to ensure his.
Please do not try to trick her out of her dog. Help her. Or let those who will help her - including health and social care services - do that.