Your DD clearly has an exceptional mind, so I think you're right to consider that what works for most children might not work for her.
My DD (who is only gifted in the 'every parent thinks their PFB is gifted' way described above) has quite a determined streak, which is probably also common in 'properly' gifted children
. I found that the usual advice to put them on the potty periodically absolutely didn't work with my DD. Backing right off and letting her own it herself was the only way to do it. We have terrible problems with food (also a control thing) and when I tried to get her to sit on the potty when she'd said she didn't need to, I could see the same behaviour starting.
Given that it's become an issue, I wonder whether your best option would be to stop for a while and let the pressure go - even though stopping and starting isn't great.
Has your DD read any potty training books? My DD liked the Pirate Pete one, and one day she put on her backpack and declared that we were going to the shop to buy a potty
Would that book be too simple for your DD to enjoy, or might she enjoy reading it herself? That might start the 'owning it' process. There's a matching princess polly one, if that's more likely to appeal.
Totally agree with the suggestion to let her decide when to do it. A friend once told me that if you can wait until they're begging you to let them do it, that's about right.
Go on a shopping expedition where she can choose the potty and 'big girl pants'. Then when you get started, just ask her occasionally whether she needs to go, but totally let her decide.
We had no wees on the potty for the first day and a half, but then we got her onto it mid-wee one time (still in pants) and somehow that made it click, and no problems from that point forward. The friend who said to wait until they begged (and did just that) had no accidents at all with her DS.