Your gran's needs are paramount and, if she is currently happy in a residential home and is not expressing a desire to return to her own home or live with one or other of her children, it would seem unneccessarily disruptive to move her particularly if she has settled well and is deriving enjoyment from being with her peers.
Does the sibling who wants to care for your gran have room for her, and do they have the resources to meet her needs 24/7? Would your gran be able to attend a day centre where she can socialise with her peers?
If they haven't experienced living with a relative who has dementia they may have no idea what they'll be taking on, or any appreciation of what being a full-time carer involves.
So many factors come in play in these situations. There may be an underlying money issue particularly if one of your gran's children believes another is seeking to control her finances/estate.
Also, people can feel immense guilt at the thought of an elderly relative going 'into a home' or being 'put away', and they may fear that others will judge them as being uncaring.
If you are to be effective as a mediator it's advisable for you to take a neutral stance, and avoid casting aspersions on the other party's competence to care either for your gran or for their children. Your sole objective should be to facilitate mutual agreement.
TBH you're between a rock and a hard place as family differences can run deep and these particular circumstances can bring out the worst, as well as the best, in hitherto harmonious families.
If you cannot bring about an accord please don't hesitate to ask your gran's advocate to intercede. This professional has an overview of your gran's current and longer term needs, and may be able to reassure other family members that your gran is in the right place and that there is no stigma to her being in a residential unit.
This isn't a simple issue of cruel v kind, or which sibling occupies the moral high ground, It's solely about providing the right care package that will meet your gran's needs and, where possible, enhance her quality of life.
It should also be borne in mind that dementia may lead your gran to exhibit increasingly challenging behaviour, and a care package that will meet her present needs may require review in the not too distant future.
I wish you the very best of luck, and I sincerely hope that your gran continues to thrive in the company of her peers.